Eigenmode twist by energy flux during fast-ion driven instabilities

Kolesnichenko, Ya. I.; Lutsenko, V. V.; Tykhyy, A. V.; Yakovenko, Yu. V.; Heidbrink, W. W.

doi:10.1063/5.0238554

Twisting of the spatial structure of eigenmodes by energy fluxes generated due to the spatial channeling (SC)—the fast-ion energy and momentum transfer across the magnetic field by destabilized modes—is considered. It is revealed that there is a correlation between the direction of the radial wave energy flux, orientation of the mode twist (MT), and the direction of mode rotation. The mode twist parameter is introduced and relations connecting it with the energy flux are established. It is shown the energy flux transforms zeros of the radial profile of the mode amplitude into minima (i.e., zeros disappear). It is found that the radial group velocity and phase velocity of reversed shear Alfvén eigenmodes (RSAEs) have opposite directions. These findings can be used for diagnostics of the energy fluxes during fast-ion driven instabilities and understanding whether the SC degrades or improves plasma performance. Specific calculations are carried out for fast magnetoacoustic modes, FMMs (known also as compressional Alfvén eigenmodes, CAEs), and Alfvén eigenmodes, AEs. A DIII-D experiment where RSAEs were observed is considered. It is concluded that an outward SC took place in this experiment. Peculiarities of various modes are discussed, which may explain why twisting of AEs, but not of FMMs, was observed experimentally in the DIII-D and NSTX tokamaks.

I. INTRODUCTION

In 2009, it was observed on the NSTX spherical tokamak that increasing the NBI heating power by a factor of three did not increase the central electron temperature but was accompanied by strong broadband Alfvénic activity; the code TRANSP predicted strong enlargement of the electron heat conductivity coefficient, $χ_{e}$ ⁠.¹ The calculated $χ_{e}$ was hardly realistic, therefore the lack of central heating is likely to be explained by the spatial channeling (SC) of the fast-ion energy by destabilized modes.² During steady-state SC, the wave energy flux $S (r)$ is directed from the driving region to the damping one. When the mode amplitude changes, the fluxes in opposite directions in different plasma regions are generated.³ Theory predicts that the SC may affect plasma performance (deteriorating or improving it) and destabilized modes by transferring both the energy and momentum.^4–8

Approximately at the same time as Ref. 2 appeared, the twist of toroidicity-induced Alfvén eigenmodes (TAEs) and reversed shear Alfvén eigenmodes (RSAEs) was observed in experiments on the NSTX and the DIII-D tokamak.⁹ This implies that the radial dependence of the mode phase, $ψ (r)$ ⁠, occurred in these experiments. A survey of these experimental results, their analysis, and the list of relevant references can be found in a recent paper,¹⁰ where, in addition, an idea to use experimental data on $ψ (r)$ for diagnostics of drive and damping was put forward. The twisting structure of destabilized high-n beta-induced Alfvén eigenmodes (BAEs) was obtained, employing the WKB-ballooning formalism, in Ref. 11. All these results suggest the presence of the wave energy flux $S (r)$ during fast-ion driven instabilities. A possible reason for the appearance of this flux is the SC, which was mentioned in Ref. 4.

A connection between the wavefront curving (and the mode twist) and the radial energy flux was established in Ref. 12. However, it is based on the assumption that the curved wavefront generates the radial energy flux, which inverts the causal relationship: it is the mode features and sources/sinks of the energy that generate the flux S and determine the wavefront. Employing this assumption, the wave parallel magnetic field, ${\tilde{B}}_{| |}$ ⁠, was artificially enlarged in Ref. 12 by adding a term with $ψ (r)$ to plasma displacement in the ideal MHD relation for ${\tilde{B}}_{| |}$ ⁠. For this reason, it is not clear whether the obtained relation for S [Eq. (10) of Ref. 12] corresponds to reality. Moreover, Eq. (10) actually implies that the mode radial group velocity is the same for any kind of mode, which is not true. Nevertheless, one cannot rule out that Eq. (10) of Ref. 12 may provide a rough estimate for the energy flux.

Thus, at present, there are no reliable relations connecting the energy flux and mode twisting, although the importance of this issue is well recognized. This fact stimulated fulfillment of the present work aimed to study the energy fluxes generated by fast-ion driven instabilities and concomitant mode twist. To achieve this aim, the eigenvalue problem in the presence of energy sources and sinks is solved. Both FMM and Alfvén eigenmodes (AEs)—global Alfvén eigenmodes (GAEs) and RSAEs—are considered.

The structure of the work is as follows. Section II contains a qualitative analysis aimed to clarify the connection between the radial energy flux generated during fast-ion driven instabilities, the eigenmode twist, and the direction of mode rotation. Sections III and IV consider energy fluxes and concomitant mode twisting during destabilization of FMMs and AEs, respectively. In addition, the work contains several appendixes. Appendix A shows how two traveling waves propagating in opposite directions produce a twisted mode. Appendix B contains derivation of a relation for the energy flux determined by energy balance. Appendix C contains derivation of energy flux in Alfvén eigenmodes. Appendix D contains relations relevant to FMMs with fast ions. Finally, Appendix E contains the coefficients of the Alfvén eigenmode equation.

II. WAVE ENERGY FLUX AND MODE TWISTING: BASIC RELATIONS

A. Energy flux

In the steady state, in the absence of SC the eigenmodes are waves standing in the radial direction but propagating along the magnetic flux surfaces. This implies that the wave energy flux across the flux surfaces vanishes,

S \cdot \nabla r = 0

⁠. In general, however,

S \cdot \nabla r \neq 0

and, therefore, the mode radial structure is phase dependent. Due to axial symmetry of the tokamak, a perturbed quantity

\tilde{X}

can be written in the form

\tilde{X} = \hat{X} (r, ϑ) exp (i ψ_{Σ} - i ω t - i n φ),

(1)

where

\hat{X} = | \tilde{X} |

⁠,

ψ_{Σ} = ψ_{Σ} (r, ϑ)

is a real quantity,

ϑ

and

φ

are the poloidal angle and toroidal angle, respectively, n is the toroidal mode number, and

ω

the mode frequency. If the mode consists of a single poloidal harmonic with poloidal mode number m (or one harmonic dominates), then

\hat{X} = \hat{X} (r)

and

ψ_{Σ} = ψ (r) + m ϑ .

(2)

Here,

ψ (r)

is the phase caused by the radial energy flux. Below we assume that Eq. is satisfied. The modes with multiple harmonics will be considered in Sec. IV.

The radial component of the Poynting vector is

S = \frac{c}{8 π} Re ({\tilde{E}}_{b}^{*} {\tilde{B}}_{| |}),

(3)

where

\tilde{E_{b}}

is the binormal component of the electric field,

{\tilde{B}}_{| |}

is the longitudinal component of the magnetic field.

In particular, for an FMM instability with small parallel wave number,

k_{| |} ≪ k

⁠, and frequencies far from ion cyclotron harmonics, we can use Eq. for

{\tilde{E}}_{b}

⁠. Neglecting the contribution of fast ions, we have

{\tilde{E}}_{b} = - i \frac{c}{ω} (i ψ^{'} p_{1} {\hat{B}}_{| |} + p_{1} {\hat{B}}_{| |}^{'} + p_{2} k_{b} {\hat{B}}_{| |}) e^{i ψ},

(4)

where

ψ

is the phase of

{\tilde{B}}_{| |}

⁠,

p_{1} = v_{A}^{2} / c^{2}

⁠,

p_{2}

is a real quantity,

k_{b}

is the binormal component of the wave vector, and “prime” denotes radial derivative. One can see that only the first term in Eq. contributes to the energy flux for which we obtain

S = κ (r) W \frac{v_{A}^{2}}{ω},

(5)

where

κ (r) \equiv ψ^{'}

⁠, and

W = {\hat{B}}_{| |}^{2} / 8 π

is the wave energy density. It follows from here that for FMM

κ / ω > 0

during outward flux (⁠

S > 0

⁠), but

κ / ω < 0

during inward flux (⁠

S < 0

⁠). In general, however, this conclusion is not true. To see this, we note that in general the mode energy flux can be written in the form

S = v_{g} W,

(6)

where

v_{g}

is the radial group velocity of the mode (not of the traveling wave), so

sgn v_{g} = sgn S

⁠. It follows from Eqs. and that in FMMs,

v_{g} = κ v_{A}^{2} / ω

⁠, and

v_{g} > 0

for

v_{p h} \equiv ω / κ > 0

⁠, where

v_{p h}

plays the role of the radial phase velocity of the mode. On the other hand, as will be shown in Sec. IV,

v_{p h} < 0

when

v_{g} > 0

in RSAEs.

Equation (5) predicts $κ \to \infty$ at the radii where $W = 0$ when $S \neq 0$ because $S \propto κ W$ ⁠. However, it is clear that the presence of zeros in the mode amplitude would prevent the radial flux. This suggests that the plasma instability leading to the energy flux modifies the mode radial structure in a way that zeros $\hat{B} = 0$ disappear; instead, $\hat{B} (r)$ is expected to have minima.

One can say, that the radial energy flux arising due to fast-ion driven instability breaks the balance between traveling waves which constitute the mode. The unbalanced part represents a traveling wave with $κ$ playing the role of the radial wave number that varies with radius.

To illustrate this, let us consider two-plane waves propagating in opposite directions. Their superposition gives

C_{1} e^{ikx + i ω t} + C_{2} e^{- ikx + i ω t} = C_{1} (2 e^{i ω t} cos k x + δ e^{- ikx + i ω t}),

(7)

where

δ = (C_{2} - C_{1}) / C_{1}

⁠. Here, the first term and second term describe a standing wave and a traveling wave, respectively. The latter transports the energy across the magnetic field. Its amplitude is much less, by a factor of

δ

⁠, than the amplitude of the standing wave because

C_{1} \approx C_{2}

due to relatively small local growth rate of the instability (⁠

γ_{L} / ω ≪ 1

⁠). Therefore, the amplitude of the mode is weakly affected everywhere except for the regions in the vicinity of the nodes

\hat{B} = 0

of the standing wave. Extending this analysis, one can show that

S \propto κ W

for any type of wave, see Appendix A.

Note that the energy flux of FMMs in the WKBJ approximation can be written in another form by replacing

κ

with the radial wave number

k_{r}

and taking

ω = k v_{A}

⁠. Then, Eq. reduces to

S_{WKBJ} = \frac{k_{r} ω}{k_{⊥}^{2}} \frac{| {\tilde{B}}_{| |} |^{2}}{8 π} .

(8)

In this form, Eq. is true also for Alfvénic perturbations, although

| {\tilde{B}}_{| |} | ≪ | \tilde{B} |

and therefore

{\tilde{B}}_{| |}

is usually neglected in studies of Alfvén waves.¹³

When the shape of a mode (FMM or AE) is weakly affected by the instability, the energy balance equation leads to the radial flux in the following form, see Appendix B:

S (r) = \frac{2}{r} \int_{0}^{r} d r^{'} r^{'} W (r^{'}) [γ_{L} (r^{'}) - γ],

(9)

where

γ = Im ω

is the instability growth rate,

γ_{L}

is the local growth/damping rate. In the steady state (⁠

γ = 0

⁠) the first term in Eq. describes the flux we refer to as

S_{s s}

(it was named

S_{heat}

in Ref. 3). It is directed from the unstable region to the damping region. When the mode amplitude grows and

S_{s s} = 0

(due to negligible damping or owing to the same radial dependence of drive and damping rates), the SC leads to redistribution of the energy within the whole region occupied by the mode. Then, Eq. describes the flux which supports the mode shape during growth of its amplitude; we refer to this flux as

S_{mode}

⁠, see details in Appendix B.

B. Mode twisting (MT)

Radial phase dependence of the mode implies twisting of the mode. In the

(r, ϑ)

-plane, the surface of constant wave phase which describes the MT degenerates to the curve

ψ (r) + m ϑ = const,

(10)

which means that

ϑ^{'} = - \frac{κ}{m},

(11)

where

ϑ^{'} = d ϑ / d r

⁠. We observe that

ϑ^{'}

is determined by

κ

(for a particular mode number). Therefore, one can refer to

κ

as a twisting parameter.

It is clear that in the

(r, φ)

-plane,

φ^{'} = \frac{κ}{n} .

(12)

Note that in the absence of the MT the level lines of $\hat{B} (r) exp (i m ϑ)$ and $\hat{B} (r) exp (i m φ)$ have non-vanishing radial derivatives, too. For this reason, experimental reconstruction, for example, of $Re \tilde{B} (r, ϑ)$ shows a joint action of $\hat{B} (r)$ and $exp (i ψ)$ on the level lines of $Re \tilde{B} (r, ϑ)$ ⁠. Therefore, we have to remind that $ϑ^{'}$ and $φ^{'}$ are defined by Eqs. (11) and (12), which do not include radial dependence of $\hat{B}$ ⁠.

To observe

ϑ^{'}

⁠, one can picture

Re \tilde{B} (r, ϑ) / \hat{B} (r) = cos (ψ + m ϑ)

⁠. Another possibility is to produce a whole picture of

Re \tilde{B} (r, ϑ)

⁠. In this case,

ϑ^{'}

can be recovered from level lines determined by Re

\tilde{B} (r, ϑ) = 0

⁠, i.e.,

ψ + m ϑ = (s + 0.5) π

with s an integer: On those lines, the

{\hat{B}}^{'}

term drops out in the equation

d Re \tilde{B} (r, ϑ) = 0

⁠, where

\begin{matrix} d Re \tilde{B} (r, ϑ) = {\hat{B}}^{'} cos (ψ + m ϑ) d r - \hat{B} sin (ψ + m ϑ) [κ d r + m d ϑ] . \end{matrix}

(13)

It follows from Eqs. (5), (11), and (12) that the mode is twisted in different directions during inward and outward energy fluxes. This fact can be used for diagnostics of the SC direction.

The main interest is the poloidal twist of the modes, which was observed experimentally.

Let us consider this issue in more details. Experimentally, it is possible to determine the direction of MT (the sign of $ϑ^{'}$ ⁠). In addition, the direction of the mode rotation can be detected. This information is sufficient to determine the direction of the radial energy flux of particular modes, i.e., the modes with known sign of the ratio $v_{g} / v_{p h}$ ⁠.

To see it, we note that it follows from Eqs. and that the mode poloidal rotation is governed by

\dot{ϑ} = \frac{ω}{m},

(14)

where

\dot{ϑ} = d ϑ / d t

⁠. Thus,

\frac{κ}{ω} = - \frac{ϑ^{'}}{\dot{ϑ}} .

(15)

In particular, for modes with

v_{g} / v_{p h} > 0

(as in FMM)

sgn S = sgn v_{p h}

⁠, and we have from Eq. that

sgn S = - sgn (ϑ^{'} \dot{ϑ}) .

(16)

Therefore, the outward/inward flux takes place when the product

θ^{'} \dot{ϑ}

is negative/positive. In particular, the outward flux (⁠

S > 0

⁠) occurs when either

ϑ^{'} > 0

⁠,

\dot{ϑ} < 0

or

ϑ^{'} < 0

⁠,

\dot{ϑ} > 0

⁠. In contrast, the inward flux (⁠

S < 0

⁠) occurs when either

ϑ^{'} < 0

⁠,

\dot{ϑ} < 0

or

ϑ^{'} > 0

⁠,

\dot{ϑ} > 0

⁠. We assume that

ω > 0

and therefore

sgn \dot{ϑ} = sgn m

⁠. Then, the modes with

m < 0

rotate in the ion diamagnetic direction [

{(B \times \nabla p_{i})}_{ϑ} < 0

] in a typical case of

p_{i}^{'} < 0

⁠, where

p_{i} (r) > 0

is the bulk ion pressure profile. For this reason they can be destabilized by the spatial inhomogeneity of the fast ions provided

ω

is less than the diamagnetic drift of these ions.

We conclude that when the direction of the MT coincides with the direction of the mode rotation, the inward energy flux of the modes with $v_{g} / v_{p h} > 0$ (FMM and GAE) takes place. Otherwise, the outward flux occurs. This is demonstrated in Figs. 1 and 2.

FIG. 1.

Sketch of |Re B̃(r,ϑ)| for |m|=3, which shows that the outward SC [ S(r)>0] takes place when the mode (FMM or GAE) is twisted in the direction opposite to that of its rotation ( ϑ̇ϑ′<0). Left panel: the twist ϑ′>0, the mode rotates in the ion diamagnetic drift direction, vdiai ( ϑ̇<0 for m<0, ω>0). Right panel: the twist ϑ′<0, the mode rotates in the electron diamagnetic drift direction, vdia(e)=−vdia(i) ( ϑ̇>0 for m>0, ω>0). Notations: white arrows show the direction of mode rotation; red arrows show the direction of ion diamagnetic drift; vdia(j) with j=i,e is the velocity of diamagnetic drift of electrons and ions.

View large Download slide

Sketch of $| Re \tilde{B} (r, ϑ) |$ for $| m | = 3$ ⁠, which shows that the outward SC [ $S (r) > 0$ ] takes place when the mode (FMM or GAE) is twisted in the direction opposite to that of its rotation (⁠ $\dot{ϑ} ϑ^{'} < 0$ ⁠). Left panel: the twist $ϑ^{'} > 0$ ⁠, the mode rotates in the ion diamagnetic drift direction, $v_{dia}^{i}$ (⁠ $\dot{ϑ} < 0$ for $m < 0$ ⁠, $ω > 0$ ⁠). Right panel: the twist $ϑ^{'} < 0$ ⁠, the mode rotates in the electron diamagnetic drift direction, $v_{dia}^{(e)} = - v_{dia}^{(i)}$ (⁠ $\dot{ϑ} > 0$ for $m > 0$ ⁠, $ω > 0$ ⁠). Notations: white arrows show the direction of mode rotation; red arrows show the direction of ion diamagnetic drift; $v_{dia}^{(j)}$ with $j = i, e$ is the velocity of diamagnetic drift of electrons and ions.

FIG. 2.

Sketch of |Re B̃(r,ϑ)| for |m|=3, which shows that the inward SC ( S(r)<0) takes place when the mode (FMM or GAE) is twisted in the direction of mode rotation ( ϑ̇ϑ′>0). Left panel: the twist ϑ′<0, the mode rotates in the ion diamagnetic drift direction, vdiai ( ϑ̇<0 for m<0, ω>0). Right panel: the twist ϑ′>0, the mode rotates in the electron diamagnetic drift direction, vdia(e)=−vdia(i) ( ϑ̇>0 for m>0, ω>0).

View large Download slide

Sketch of $| Re \tilde{B} (r, ϑ) |$ for $| m | = 3$ ⁠, which shows that the inward SC (⁠ $S (r) < 0$ ⁠) takes place when the mode (FMM or GAE) is twisted in the direction of mode rotation (⁠ $\dot{ϑ} ϑ^{'} > 0$ ⁠). Left panel: the twist $ϑ^{'} < 0$ ⁠, the mode rotates in the ion diamagnetic drift direction, $v_{dia}^{i}$ (⁠ $\dot{ϑ} < 0$ for $m < 0$ ⁠, $ω > 0$ ⁠). Right panel: the twist $ϑ^{'} > 0$ ⁠, the mode rotates in the electron diamagnetic drift direction, $v_{dia}^{(e)} = - v_{dia}^{(i)}$ (⁠ $\dot{ϑ} > 0$ for $m > 0$ ⁠, $ω > 0$ ⁠).

For modes with $v_{g} / v_{p h} < 0$ (RSAE), Figs. 1 and 2 are relevant to the inward SC and outward SC, respectively.

C. Persistence of the modes

In the analysis above, we assumed that the SC does not change the mode frequency and the mode radial structure except for the regions where the mode amplitude in the absence of SC has zeros. This is a reasonable approximation when the energy flux of the mode, S, is much less than the energy flux of traveling waves constituting the mode,

S^{t}

⁠,

S ≪ S^{t} .

(17)

This implies that the group velocity of traveling waves should be large enough to satisfy the condition^4,7

v_{g}^{t} ≫ γ_{α} Δ_{α},

(18)

where

γ_{α}

is the local instability growth rate,

Δ_{α}

is the width of the region where drive dominates. For FMM,

v_{g}^{t} = k_{r} v_{A} / k

⁠. For AEs, the relations for the mode group velocity are more complicated, see Sec. IV.

III. FAST MAGNETOACOUSTIC MODES

A. A model for description of FMM instability

We proceed from an equation for

{\tilde{B}}_{| |}

⁠, which we obtain from Eq. (D44) (which corresponds to the FMM equation in Ref. 14). Neglecting the term with

p_{2}

(see Appendix D) and introducing the function

ν (r)

to model the destabilizing influence of fast ions and wave damping, which we treat as a parameter, we write

\frac{c^{2}}{ω^{2}} \frac{1}{r} \frac{d}{d r} r p_{1} \frac{d {\tilde{B}}_{| |}}{d r} + (1 - \frac{c^{2} k_{b}^{2}}{ω^{2}} p_{1}) {\tilde{B}}_{| |} = 2 i ν (r) {\tilde{B}}_{| |},

(19)

where

p_{1} = v_{A}^{2} / c^{2}

for both

ω^{2} ≪ ω_{B i}^{2}

and

ω^{2} ≫ ω_{B i}^{2}

⁠,

ω \neq l ω_{B i}

⁠, l is an integer.

The function

ν (r)

has a simple physical meaning. To see it, let us multiply Eq. by

r {\tilde{B}}_{| |}^{*} ω^{2} / c^{2}

(the subscript “*” means complex conjugate value) and integrate the product over the radial coordinate,

\begin{matrix} \int_{0}^{\infty} d r r [- p_{1} | {\tilde{B}}_{| |}^{'} |^{2} + (\frac{ω^{2}}{c^{2}} - k_{b}^{2} p_{1}) | {\tilde{B}}_{| |} |^{2}] = 2 i \frac{ω^{2}}{c^{2}} \int_{0}^{\infty} d r r | {\tilde{B}}_{| |} |^{2} ν . \end{matrix}

(20)

Writing

ω = ω_{0} + i γ

⁠, with

γ ≪ ω_{0}

the instability growth rate, we find (the subscript “0” is omitted)

\frac{γ}{ω} = \frac{\int_{0}^{\infty} d r r ν | {\tilde{B}}_{| |} |^{2}}{\int_{0}^{\infty} d r r | {\tilde{B}}_{| |} |^{2}} .

(21)

It follows from this that

ν (r)

represents the ratio of the local growth/damping rate

γ_{L}

to the mode frequency,

ν = γ_{L} / ω .

(22)

When the instability arises due to anti-Hermitian part of the dielectric tensor component

ϵ_{1}

⁠,

p_{1} = Re p_{1} (1 - i \frac{v_{A}^{2}}{c^{2}} Im ϵ_{1}) .

(23)

Then, Eq. with

ν = 0

yields

\frac{γ}{ω} = - \frac{\int_{0}^{\infty} d r r (| {\tilde{B}}_{| |}^{'} |^{2} + k_{b}^{2} | {\tilde{B}}_{| |} |^{2}) \frac{v_{A}^{2}}{2 ω^{2}} Re P Im ϵ_{1}}{\int_{0}^{\infty} d r r | {\tilde{B}}_{| |} |^{2}} .

(24)

We observe that the integrand in the numerator contains the term with the radial derivative of the mode amplitude. It can be neglected provided

| {\tilde{B}}_{| |}^{'} |^{2} ≪ k_{b}^{2} | {\tilde{B}}_{| |} |^{2}

⁠, which implies that the poloidal mode number (m) is sufficiently high. Then, Eq. reduces to

\frac{γ}{ω} = - \frac{\int_{0}^{\infty} d r r | {\tilde{B}}_{| |} |^{2} \frac{k_{b}^{2} v_{A}^{2}}{ω^{2}} \frac{v_{A}^{2}}{2 c^{2}} Im ϵ_{1}}{\int_{0}^{\infty} d r r | {\tilde{B}}_{| |} |^{2}} .

(25)

Comparing Eqs. and , we conclude that in the considered case

ν = - \frac{k_{b}^{2} v_{A}^{2}}{ω^{2}} \frac{v_{A}^{2}}{2 c^{2}} Im ϵ_{1} .

(26)

Plasma is on the margin of stability (⁠

γ = 0

⁠) provided

\int_{0}^{\infty} d r r {\hat{B}}_{| |}^{2} ν (r) = 0,

(27)

which follows from Eq. . Regions where

\hat{B}

is close to zero weakly contribute to Eq. and to

S (r)

⁠. On the other hand, these regions play an important role in the mode twisting because

ϑ^{'} \propto κ

⁠,

S \propto κ \hat{B}

⁠, and

S (r)

is a continuous function.

B. Twisting parameter $κ$

Proceeding from Eq. and using Eqs. and , we obtain the following equation:

\begin{matrix} \frac{1}{v_{A}^{2} r} \frac{d}{d r} r v_{A}^{2} \frac{d {\hat{B}}_{| |}}{d r} + (\frac{ω^{2}}{v_{A}^{2}} - \frac{m^{2}}{r^{2}} - κ^{2}) {\hat{B}}_{| |} \\ = - i [(κ^{'} + \frac{{(r v_{A}^{2})}^{'}}{r v_{A}^{2}} κ - \frac{2 ν ω^{2}}{v_{A}^{2}}) {\hat{B}}_{| |} + 2 κ {\hat{B}}_{| |}^{'}], \end{matrix}

(28)

where “prime” denotes

d / d r

⁠. Because

{\hat{B}}_{| |}

and

κ

are real quantities, Eq. splits into two equations,

\frac{1}{v_{A}^{2} r} \frac{d}{d r} r v_{A}^{2} \frac{d {\hat{B}}_{| |}}{d r} + (\frac{ω^{2}}{v_{A}^{2}} - \frac{m^{2}}{r^{2}} - κ^{2} (r, {\hat{B}}_{| |})) {\hat{B}}_{| |} = 0,

(29)

[κ^{'} + \frac{{(r v_{A}^{2})}^{'}}{r v_{A}^{2}} κ - \frac{2 ν ω^{2}}{v_{A}^{2}}] {\hat{B}}_{| |} + 2 κ {\hat{B}}_{| |}^{'} = 0.

(30)

The first of these equations is well known when

ν = 0

⁠, in which case it determines the FMM amplitude in stable plasmas. Its solution for

m ≫ 1

was obtained in Ref. 15, and it was analyzed in detail in Ref. 14. When

v_{A} (r) = const

⁠, Bessel functions of the mth order satisfy this equation. The presence of

κ^{2}

in Eq. should modify

\hat{B}

in a way that its zeros disappear. The second equation with

ν = 0

is satisfied by

κ (r) = 0

⁠, and

κ

remains small when

ν ≪ 1

⁠. Therefore, we can neglect

κ^{2}

in Eq. assuming that the instability drive is weak (⁠

ν ≪ 1

⁠) and

κ^{2} ≪ m^{2} / r^{2}

(31)

everywhere within the mode width except for the narrow regions around the radii where

{\hat{B}}_{| |} = 0

determined by Eq. with

κ = 0

⁠.

Equation can be considered as an equation for

κ

with given

\hat{B}

⁠. Therefore, we write it as

κ^{'} + {[ln (r {\hat{B}}_{| |}^{2} v_{A}^{2})]}^{'} κ = 2 ν \frac{ω^{2}}{v_{A}^{2}} .

(32)

For the boundary condition, we take

κ (0) = 0

⁠, which is a consequence of

S (0) = 0

given

{\hat{B}}_{| |} (0) = 0

[the

m = 0

FMM having

{\hat{B}}_{| |} (0) \neq 0

is eliminated from consideration by condition Eq. ]. Then, the solution of Eq. is

κ (r) = \frac{2 ω^{2}}{r {\hat{B}}_{| |}^{2} (r) v_{A}^{2} (r)} \int_{0}^{r} d y y ν (y) {\hat{B}}_{| |}^{2} (y) .

(33)

This relation for

κ

and Eq. lead to the energy flux in the form of Eq. with

γ = 0

⁠.

C. Specific examples

1. Zero width of drive/damping regions

Let us assume that the regions of drive and damping are strongly localized and consider two cases: first,

S = S_{s s}

due to

γ = 0

and, second, when

S = S_{mode}

due to

γ_{L} (r) > 0

⁠.

The integral in Eq. (33) can be calculated without finding $\hat{B} (r)$ by taking

$ν (r) = \sum_{j} a ν_{j} δ (r - r_{j}) = a ν_{+} δ (r - r_{+}) + a ν_{-} δ (r - r_{-}),$

(34)

where $δ (r - r_{j})$ is the Dirac delta function, $ν_{\pm} = γ_{\pm} / ω$ with $γ_{\pm} = γ_{L} (r_{\pm})$ ⁠; the subscripts “+” and “−” label the radius of drive and damping, respectively $(ν_{+} > 0, ν_{-} < 0)$ ⁠, a is the plasma radius. Then, $κ (r) = 0$ for $r < min (r_{+}, r_{-})$ and $r > max (r_{+}, r_{-})$ ⁠. In the region between drive and damping $κ (r) \neq 0$ ⁠: For outward/inward SC,

$κ (r) = \frac{2 a ω γ_{\pm}}{v_{A}^{2} (r)} \frac{r_{\pm} {\hat{B}}_{\pm}^{2}}{r {\hat{B}}_{| |}^{2} (r)},$

(35)

where ${\hat{B}}_{\pm} = {\hat{B}}_{| |} (r_{\pm})$ ⁠. Thus, $κ > 0$ for the outward SC and $κ < 0$ for the inward SC, which agrees with Eq. (5); $| κ |$ is the same for both cases. This twisting parameter is convenient to write in dimensionless variables as follows:

$\bar{κ} = 2 \bar{ω} ν_{\pm} η (x) \frac{x_{\pm} W_{\pm}}{x W (x)},$

(36)

where $\bar{κ} = a κ$ ⁠, $\bar{ω} = ω a / v_{A 0}$ ⁠, $v_{A 0} = v_{A} (0)$ ⁠, $η = n_{i} (x) / n_{i} (0)$ ⁠, $x = r / a$ ⁠, $W_{\pm} = W (r_{\pm})$ ⁠. The phase $ψ$ is determined by

$ψ (r) = \int_{0}^{r} κ (r_{1}) d r_{1} + ψ_{0},$

(37)

where $ψ_{0}$ can be taken zero. Then, for $ν (r)$ given by Eq. (34), $ψ (r) \neq const$ only in the region between the radii of drive and damping, $min (r_{+}, r_{-}) < r < max (r_{+}, r_{-})$ ⁠. The energy flux we obtain by substituting $κ$ given by Eq. (35) into Eq. (5),

$| S | = | S_{s s} | = 2 a γ_{\pm} W_{\pm} \frac{r_{\pm}}{r}, S_{mode} = 0.$

(38)

Thus, $| S (r) | \propto 1 / r$ in the region $min (r_{+}, r_{-}) < r < max (r_{+}, r_{-})$ ⁠, this radial dependence results from cylindrical geometry.
Let us assume now that $ν_{-} = 0$ to but $γ \neq 0$ to observe $S_{mode}$ when $S_{s s} = 0$ ⁠. In this case, Eq. (9) leads to inward flux in the region $r < r_{+}$ ⁠, c.f.,³

$S_{mode} (r < r_{+}) \equiv S_{1} = - 2 γ \frac{1}{r} \int_{0}^{r} d y y W .$

(39)

One can see that

| r S_{1} (r) |

is a growing function, vanishing at

r = 0

and maximum at

r = r_{1}

⁠. In the region

r > r_{+}

⁠,

S_{mode} (r > r_{+}) \equiv S_{2} = 2 a γ_{+} W_{+} \frac{r_{+}}{r} + S_{1} .

(40)

Because

S (\infty) = 0

⁠,

S_{2}

represents the outward flux. One can show that

r S_{2}

is a decreasing function with

S_{2} (r_{+} + 0) = | S_{1} (r_{+} - 0) |

⁠.

2. Regions of drive and damping are well separated

Numerical calculations are required to take into account the finite width of the driving/damping region. We use Eq. written in the form

\frac{η (x)}{x} \frac{d}{d x} \frac{x}{η (x)} \frac{d {\tilde{B}}_{| |}}{d x} + ({\bar{ω}}^{2} η (x) - \frac{m^{2}}{x^{2}} - 2 i ν {\bar{ω}}^{2} η (x)) {\tilde{B}}_{| |} = 0.

(41)

In contrast to Eq. employing variables

\hat{B}

and

ψ

⁠, below we use

B_{1} = Re \tilde{B}

and

B_{2} = Im {\tilde{B}}_{| |}

⁠. In this case,

{\hat{B}}^{2} = B_{1}^{2} + B_{2}^{2}

⁠,

ψ = arctan (B_{1} / B_{2})

⁠, and

\bar{κ} \equiv a κ = \frac{d ψ}{d x} = \frac{2 η (x) {\bar{ω}}^{2}}{x {\hat{B}}_{| |}^{2} (x)} \int_{0}^{x} d y y ν (y) {\hat{B}}_{| |}^{2} (y) .

(42)

Instead of Eq. , we write

ν (x) = \frac{ν_{+}}{\sqrt{π} X} exp [- \frac{{(x - x_{+})}^{2}}{X^{2}}] + \frac{ν_{-}}{\sqrt{π} X} exp [- \frac{{(x - x_{-})}^{2}}{X^{2}}],

(43)

where X is small; we will take

X = 0.025

⁠.

To demonstrate the effects of the instability on the mode twist, we use a simple model with ${\tilde{B}}_{| |} = 0$ at the plasma edge and homogeneous plasma. In this case, the solution of Eq. (41) in the absence of instability is ${\hat{B}}_{| |} = J_{m} (ξ_{s} x)$ ⁠, where $J_{m} (ξ_{s} x)$ is the Bessel function of the first kind of the mth order, s is an integer, and $ξ_{s}$ are zeros of the Bessel function. We selected the solution with $s = 0$ ⁠, which has no nodes (no zeros at $0 < x < 1$ ⁠). Two cases are considered: first, the steady state; second, growth of the mode amplitude with negligible wave damping. Calculations are made for a mode with $m = - 3$ ⁠.

Calculated energy fluxes are shown in Fig. 3. We observe that the flux $S_{s s}$ is directed outward, whereas $S_{mode}$ is directed inward/outward to the right/left of the $r_{+}$ radius, as expected. The flux $S_{s s}$ exceeds $S_{mode}$ when $γ / ω < 0.1$ ⁠.

FIG. 3.

Energy fluxes of the s=0 mode (the mode without nodes) driven by ν(x)≡γL/ω given by Eq. (43): (i) Red curve, Sss in the steady state, γ=0 when ν+=0.05, x+=0.4, and ν−=0.0291, x−=0.9; the damping was selected in a way to provide γ=0. (ii) Black curves, Smode with ν−=0 for various magnitudes of γ when x+=2/3.

View large Download slide

Energy fluxes of the $s = 0$ mode (the mode without nodes) driven by $ν (x) \equiv γ_{L} / ω$ given by Eq. (43): (i) Red curve, $S_{s s}$ in the steady state, $γ = 0$ when $ν_{+} = 0.05$ ⁠, $x_{+} = 0.4$ ⁠, and $ν_{-} = 0.0291$ ⁠, $x_{-} = 0.9$ ⁠; the damping was selected in a way to provide $γ = 0$ ⁠. (ii) Black curves, $S_{mode}$ with $ν_{-} = 0$ for various magnitudes of $γ$ when $x_{+} = 2 / 3$ ⁠.

Figure 4 shows the mode structure in the $(r, ϑ)$ plane—i.e., $\hat{B} (r) | cos (ψ + m ϑ) |$ —and the value of $cos (ψ + i m ϑ)$ during SC. The latter eliminates the influence of the radial dependence of the mode amplitude on $ϑ^{'}$ and provides $ϑ^{'}$ ⁠. Thanks to this, one can observe the direction of the MT even in the case when it is so small that the level lines of $ϑ (r)$ in the presence of SC almost coincide with those in the absence of SC.

FIG. 4.

Structure of an FMM with m=−3 in the ( r,ϑ)-plane during outward SC when ν(x) is given by Eq. (43) with the same parameters as in Fig. 3: Upper panel, B̂(r)| cos(ψ+mϑ)|; lower panel, cos(ψ+mϑ). White dashed curves represent level lines of B̂(r)| cos(ψ+mϑ)| in the absence of instability.

View large Download slide

Structure of an FMM with $m = - 3$ in the (⁠ $r, ϑ$ ⁠)-plane during outward SC when $ν (x)$ is given by Eq. (43) with the same parameters as in Fig. 3: Upper panel, $\hat{B} (r) | cos (ψ + m ϑ) |$ ⁠; lower panel, $cos (ψ + m ϑ)$ ⁠. White dashed curves represent level lines of $\hat{B} (r) | cos (ψ + m ϑ) |$ in the absence of instability.

Experimentally, the picture of $cos (ψ + m ϑ)$ can be produced when, first, $\hat{B} (r) cos (ψ + m ϑ)$ is known (for instance, due to tomographical reconstruction of soft x-ray measurements, e.g., as was done for an Wendelstein 7-AS experiment in Ref. 16) and, second, $\hat{B} (r)$ is determined (see, e.g., Ref. 17). Then, the energy flux $S (r)$ can be calculated by means of Eq. (5) using $κ = - ϑ^{'} / m$ ⁠, where $ϑ^{'}$ is recovered from the picture of $cos (ψ + m ϑ)$ ⁠.

3. Continuous transition from drive to damping

To investigate SC when drive and damping regions are wide, Eq. (41) was solved with complex ${\tilde{B}}_{| |} (x)$ and a smooth $ν (x)$ ⁠. The boundary conditions were ${\tilde{B}}_{| |} = const \cdot x^{m}$ at $x \to 0$ and ${\tilde{B}}_{| |} (1) = 0$ ⁠. Mode amplitude was normalized such that $max | {\tilde{B}}_{| |} (x) | = 1$ ⁠. The form $ν (x) = C (d - x)$ was chosen, with C and d being constants. The normalized density profile was taken in the form $η (x) = {(1 - x^{2})}^{α}$ with $α = const$ ⁠. The value of d was adjusted for every set of parameters (m, number of radial nodes s, $α$ ⁠, C) so that the mode was on the margin of stability, $Im \bar{ω} = 0$ ⁠, and accordingly no energy flux would be diverted into the growth of the mode amplitude.³ Calculations were performed for density profiles ranging from uniform (⁠ $α = 0$ ⁠) to peaked (⁠ $α = 2$ ⁠). Smooth dependence of mode frequency and localization on $α$ was observed for $0 \leq C \leq 1$ ⁠, with $\bar{ω}$ growing almost linearly with increasing $α$ and the mode shifting inward, as expected from theory. As no important difference was observed in the behavior of modes depending on $α$ ⁠, the results for $α = 1 / 2$ [ $η (x) = {(1 - x^{2})}^{\frac{1}{2}}$ ] are given below.

For every m, a series of modes was found in the absence of drive/damping (⁠ $C = 0$ ⁠) with different number s of radial nodes ${\tilde{B}}_{| |} = 0$ ⁠. As expected from comparison of Eq. (41) with the Bessel equation, the mode frequency increases with s. Modes with $s = 0$ and the lowest $\bar{ω}$ are characterized by smooth dependence of $κ$ on radius (see Fig. 5). SC barely changes $\bar{ω}$ and the radial mode structure even for strong local drive/damping (⁠ $C \sim 1$ ⁠). As the strength of SC increases, the mode twists and acquires a spiral shape.

FIG. 5.

View large Download slide

FMM with poloidal mode number $m = - 3$ and no radial nodes (node number $s = 0$ ⁠) with density profile $n_{i} \sim \sqrt{1 - {(r / a)}^{2}}$ and drive/damping strength parameter $C = 0.6$ ⁠.

The influence of energy flux on the mode structure is much more pronounced for modes with $s \neq 0$ ⁠. These modes persist when $C \neq 0$ ⁠, but zeros of ${\tilde{B}}_{| |} = 0$ at radial nodes disappear and transform into minima. The twisting parameter $κ = ψ^{'}$ is discontinuous in the absence of drive/damping (the phase $ψ$ jumps by $\pm π$ at every radial node as ${\tilde{B}}_{| |}$ changes sign). These discontinuities, as well as the minima of ${\hat{B}}_{| |}$ at radial nodes, are progressively smoothed out as C increases [compare Figs. 6(b) and 6(c)]. In the poloidal cross section, one can see that for small values of C the energy flux connects local maxima of ${\hat{B}}_{| |}$ which are adjacent by radius x and separated by $π / m$ in $ϑ$ [see Fig. 6(a)]. A comparison with an FMM with the same $m = - 3$ but no radial nodes (⁠ $s = 0$ ⁠) (Fig. 5) shows that mode twisting strongly depends on s: the more localized $s = 0$ mode is barely twisted even at $C = 0.6$ despite close values of drive/damping strength $| γ_{L} / ω | ≳ 0.1$ at points where drive/damping is most strongly coupled with the mode (i.e., where $| {\hat{B}}_{| |} γ_{L} / ω |$ is maximum). Mode twisting for a given drive/damping strength is stronger for lower values of m, as expected from Eq. (11).

FIG. 6.

View large Download slide

Same as Fig. 5 but with two nodes, $s = 2$ ⁠.

The frequency of the mode is weakly affected by SC, especially for higher s; e.g., for the mode shown in Fig. 6, $(\bar{ω} - {\bar{ω}}_{0}) / {\bar{ω}}_{0} \sim 10^{- 3}$ ⁠, where ${\bar{ω}}_{0}$ is the frequency of the undriven mode. Calculations indicate that unlike TAEs, which are destroyed by strong SC,⁴ FMMs persist even for unrealistically large values of $γ_{L} / ω \sim 0.25$ ⁠; for the same $m = - 3$ ⁠, $s = 2$ mode, such strong SC changes the mode frequency only by $2.5 %$ ⁠.

For modes with $s \neq 0$ and $C \neq 0$ ⁠, the maximum value of $κ$ ⁠, which occurs at the minimum of mode amplitude, the value of normalized ${\hat{B}}_{| |}$ at the minimum, and the maximum value of S are found to depend smoothly on C: $max κ (x) \sim C^{- 1}$ ⁠, $min {\hat{B}}_{| |} \sim C$ ⁠, and $max S \sim C$ as expected from $S \sim κ {\hat{B}}_{| |}^{2}$ ⁠. The maximum value of $κ$ for a mode with given drive/damping strength parameter C also increases with s, indicating that $κ$ may be considered as an equivalent to radial wave number.

Note that for modes with $m \neq 0$ ⁠, the values of $γ_{L} / ω$ relevant to a given mode are much smaller than C, as for the chosen form of $ν (x)$ ⁠, $| ν (x) |$ is maximum at $x = 0$ and $x = 1$ ⁠, where the mode amplitude is zero due to boundary conditions.

IV. ALFVÉN EIGENMODES

A. Global Alfvén eigenmodes

When studying the GAEs,^18,19 we use reduced MHD equations presented in Appendix E, describing Alfvén eigenmodes in terms of the function

ϕ

⁠, where

\partial ϕ / \partial t = \tilde{Φ}

⁠,

Φ

is the mode scalar potential. We neglect all factors causing the mode coupling (toroidicity, Shafranov shift and cross section shaping) and consider a single wave harmonic in the cylindrical geometry,

ϕ = R e [ϕ_{m n} (r) exp (i m ϑ - i n φ - i ω t)] .

(44)

Then, the matrices entering the Alfvén eigenmode equation, Eq. , become scalar. Putting

ε = Δ^{'} = 0

into their expressions given in Appendix E, we obtain the following equation for the radial profile of a single wave harmonic,

ϕ_{m n} (r)

⁠:

\begin{matrix} {\tilde{ω}}^{2} [\frac{m^{2} η}{r} ϕ_{m n} - \frac{d}{d r} (r η \frac{d ϕ_{m n}}{d r})] \\ = (\frac{m^{2} k_{m n}^{2}}{r} - 2 m k_{m n} ι^{'}) ϕ_{m n} - \frac{d}{d r} (r k_{m n}^{2} \frac{d ϕ_{m n}}{d r}) + i \tilde{ω} ν (r) ϕ_{m n}, \end{matrix}

(45)

where

\tilde{ω} = ω / ω_{A 0}

⁠,

ω_{A 0} = v_{A} (r = 0) / R

⁠,

k_{m n} = R k_{∥} = m ι - n

⁠,

ι = 1 / q

⁠, q is the magnetic winding number (safety factor), R is the plasma major radius,

ν (r)

characterizes the mode drive and damping.

The radial energy flux given by Eq. becomes

S = \frac{c^{2} ω}{8 π} (k_{∥}^{2} - \frac{ω^{2}}{v_{A}^{2}}) Im (ϕ_{m n}^{*} ϕ_{m n}^{'}) .

(46)

Taking into account that

κ = {| ϕ |}^{- 2} Im (ϕ_{m n}^{*} ϕ_{m n}^{'}),

(47)

which follows from Eq. , we obtain

S = \frac{c^{2} ω}{8 π} (k_{∥}^{2} - \frac{ω^{2}}{v_{A}^{2}}) | ϕ |^{2} κ .

(48)

This equation does not utilize the WKBJ approximation. Admittedly, it may require certain corrections if the contribution of satellite harmonics to the mode is strong.

As the GAE group velocity is¹³

v_{g}^{t} = \frac{v_{A}^{2}}{ω k_{⊥}^{2}} (k_{∥}^{2} - \frac{ω^{2}}{v_{A}^{2}}) k_{r}

(49)

and the energy density is

W = \frac{c^{2}}{v_{A}^{2}} \frac{ω^{2} k_{⊥}^{2} | ϕ |^{2}}{8 π},

(50)

Eq. agrees with Eq. obtained within the WKBJ approximation.

Let us estimate the ratio

S_{r} / (W κ) = v_{g} / κ

of the lowest-frequency GAE (i.e., the mode without radial nodes) near the Alfvén continuum minimum,

r = r_{*}

⁠, which approximately coincides with the mode amplitude maximum. We assume that

g ≫ 2

⁠, where

g = - {(\frac{2 ω_{A}^{2}}{r {(ω_{A}^{2})}^{″}} \frac{η^{'}}{η}) |}_{r = r_{*}},

(51)

ω_{A} = k_{∥} v_{A}

⁠, and take

k_{∥}^{2} = \frac{ω^{2}}{v_{A}^{2}} (1 + τ),

(52)

at

r = r_{*}

⁠, where

τ

is expected to be small. Then, we get from Eqs. and that

\frac{v_{g}}{κ} = \frac{v_{A}^{2}}{ω} \frac{k_{∥}^{2}}{k_{⊥}^{2}} τ .

(53)

After straightforward calculations, we obtain from Eqs. – of Ref. 13

τ \approx \frac{\sqrt{g} - 1}{\sqrt{g}} \frac{r_{*}}{m^{2} L_{η}},

(54)

where

L_{η} = η (r_{*}) / η^{'} (r_{*})

⁠. We observe that

τ ≲ (0.3 \div 1) / m^{2}

when

L_{η} \sim 1

⁠.

B. Reversed shear Alfvén eigenmodes

Although a RSAE²⁰ is known to have one main Fourier harmonic that dominates the rest, its existence is made possible by the toroidal satellites of this harmonic (or it exists as an Energetic Particle Mode, EPM),^21–23 which complicates analytical treatment of the RSAE twist. For this reason, the RSAE twist is studied here numerically with a dedicated code. The code solves Eq. (C13), utilizing Hermitian cubic finite elements.

We use

δ

-like radial profiles of drive and damping, with drive and damping concentrated in two narrow regions apart from the mode maximum. In this case, we observe a shelf-like energy flux profile, which is convenient for both controlling the accuracy of the code and establishing the relation between the energy flux and the mode twist. In reality, the AE damping is often radially localized when it takes place via continuum damping. Although in our calculations the eigenmodes often manifest strong interaction with continuum (localized spikes on the mode radial profile), we put both damping and drive regions between the mode tip and the continuum resonances. The reason is that it is difficult to simulate the continuum damping realistically within ideal MHD. At the same time, it does not seem that the specific physics of the damping mechanism is important for us. We take the matrix coefficient

ν (x)

in the following form:

ν_{i j} (x) = {\begin{array}{l} ν_{+} h (x, x_{+}, X_{+}) & if i = j = i_{+}, \\ ν_{-} h (x, x_{+}, X_{+}) & if i = j = i_{-}, \\ 0 & otherwise, \end{array}

(55)

h (x, x_{0}, X) = \frac{1}{2 X} (\frac{1}{4} {| \frac{x - x_{0}}{X} |}^{3} - \frac{3}{4} {| \frac{x - x_{0}}{X} |}^{2} + 1),

(56)

where

ν_{+}

and

ν_{-}

determine the intensities of drive and damping, respectively,

i_{+, -}

are the indices of the wave harmonics affected by these processes,

x_{+, -}

and

X_{+, -}

are the radial localizations and widths, respectively, of the drive and damping zones. Note that when

i_{+} = i_{-}

⁠, the first two lines of Eq. should be united in an obvious way. Equation can be considered as a generalization of Eq. . The differences are, first, that here

ν (x)

is a Hermitian matrix function and, second, that the bell-like function has finite support. Due to this, we can arrange the drive so that it does not reach the Alfvén resonance points, where the wave function is singular.

We use an analytically derived approximation for the matrices $A^{j k}$ and $D^{j k}$ ⁠, $j, k = 0, 1$ ⁠, which are valid for plasma equilibrium with large aspect ratio of the plasma torus and quasicircular flux-surface cross sections (see Appendix E).

The calculations were carried out for deuterium plasmas in a tokamak with $B (r = 0) = 2.2 T$ ⁠, $R = 167 cm$ ⁠, $a = 35 cm$ ⁠. The profiles of electron density (⁠ $n_{e}$ ⁠) and temperature (T) were taken in the form $f (r) = f_{0} {(1 - α x^{2})}^{ς}$ with $n_{e 0} = 3 \times 10^{13} {cm}^{- 3}$ ⁠, $T_{0} = 2.15 keV$ ⁠, $ς = 0.5$ for density, and $ς = 1$ for temperature. The parameters $α$ in the density and temperature profiles were chosen so that $n_{e} (r = a) = 0.1 n_{e 0}$ ⁠, $T (r = a) = 0.05 T_{0}$ ⁠.

Figures 7–10 show an example of two RSAEs calculated with balanced drive and damping. One of the modes has no radial nodes (⁠ $s = 0$ ⁠); the other one has one radial node (⁠ $s = 1$ ⁠).

FIG. 7.

View large Download slide

Nodeless RSAE (⁠ $s = 0$ ⁠) with the dominant harmonic numbers $m = - 12$ ⁠, $n = - 4$ during outward SC.

FIG. 8.

View large Download slide

Same as Fig. 7 but for the mode with $s = 1$ ⁠.

FIG. 9.

View large Download slide

q-profile and Alfvén continuum corresponding to Figs. 7 and 8.

FIG. 10.

View large Download slide

Radial profiles of $γ_{L}$ ⁠, energy flux, and mode twist. Notations: red lines correspond to the mode with $s = 0$ ⁠; blue lines, with $s = 1$ ⁠.

In our simulation, the code iteratively adjusted

ν_{+}

and

ν_{-}

in order to minimize

\tilde{γ} = Im \tilde{ω}

for each mode separately. At each iteration, the code found the eigenfunction, calculated

γ_{L}

employing Eq. , and adjusted the non-zero elements of

ν_{+}

and

ν_{-}

so that the mode growth rate would be close to zero. The growth rate was evaluated via the relationship

γ = \int_{x_{l}}^{x_{r}} d x x γ_{L} (x) W (x) / \int_{x_{l}}^{x_{r}} d x x W (x) \approx 0,

(57)

where

x_{l}

and

x_{r}

were chosen so that the integration interval included the region where the mode is large but not the points of Alfvén resonance, where the mode energy density

W (x)

is singular. When we took

x_{l} = 0

and

x_{r} = 1

⁠, the integral was affected by random grid effect at the resonance points. It seems that the effect of the Alfvén continuum (AC) on the mode energy balance deserves more attention in the future.

The mode frequencies, $\tilde{ω} \approx 0.144 + 5.54 \times 10^{- 5} i$ and $0.137 + 5.85 \times 10^{- 5} i$ ⁠, lie near the TAE gap. The modes interact with AC at $x > 0.8$ [see Fig. 9(b)], which manifests itself as spikes on graphs in Figs. 7(b) and 8(b) and in irregular behavior of node lines in Figs. 7(a) and 8(a). The mode maxima are at $x_{0} \approx 0.59 - 0.595$ ⁠, near the minimum of q, which is at $x_{*} = 0.6$ [see Fig. 9(a)].

Drive and damping were applied to the dominant Fourier harmonic,

(m, n) = (- 12, - 4)

⁠, at

x_{+} = 0.535

and

x_{-} = 0.645

⁠, respectively (the energy flux was directed outward). The widths of the drive and damping zones were

X_{+} = X_{-} = 0.02

⁠. The local drive/damping rate of the modes [see Eq. ] are shown in Fig. 10(a). As mentioned above, the drive and the damping are balanced during the calculations, so both modes are close to steady state,

γ / ω \sim 4 \times 10^{- 4}

⁠. The drive and the damping taken separately can be characterized by partial drive/damping rates,

- \tilde{γ} [x_{0}, x_{r}] \approx \tilde{γ} [x_{l}, x_{0}] \approx 1.5 \times 10^{- 3} \sim 10^{- 2} \tilde{ω}

for the

s = 0

mode and

3.7 \times 10^{- 4} \sim 3 \times 10^{- 3} \tilde{ω}

for the

s = 1

mode, where

\tilde{γ} [x_{1}, x_{2}] = \int_{x_{1}}^{x_{2}} d x x \frac{γ_{L} (x)}{ω_{A 0}} W (x) / \int_{0}^{1} d x x W (x) .

(58)

To exclude the continuum influence, we took

x_{l} = 0.4

and

x_{r} = 0.7

in our calculations. We observe that the mode twist is quite observable at rather weak mode drive.

The radial energy flux and the twisting parameter $κ$ calculated via Eqs. (C15) and (C20), respectively, are shown in Fig. 10(b). In the interval between the drive and damping regions, the flux behaves as ${\bar{S}}_{r} \propto 1 / r$ ⁠, as expected from conservation of energy. We observe that the $s = 1$ mode [blue lines in Fig. 10(b)] is characterized by a smaller energy flux but a stronger twist. The twist of the $s = 1$ mode is especially strong near the node, but it exceeds the twist of the $s = 0$ mode everywhere. It seems that the energy flux transported by the mode decreases as the mode frequency approaches the continuum, as is the case for the GAEs [see Eq. (48)].

Calculations for different balanced drive and damping rates show that $κ \propto \bar{S}$ ⁠, although there is some deviation from this proportionality law at strong drives, when the mode profile becomes distorted.

We observe in Fig. 10(b) that $κ < 0$ although $\bar{S} > 0$ ⁠. This means that $k_{r} < 0$ and the wave phase velocity $v_{p h}^{t} = ω / k_{r} < 0$ (assume $ω > 0$ ⁠) dominates in traveling waves constituting the mode. On the other hand, the relation $S = v_{g} W \propto κ W$ implies $v_{g} > 0$ when $S > 0$ ⁠, so $v_{g} v_{g}^{t} < 0$ ⁠, $v_{g}^{t} v_{p h}^{t} < 0$ ⁠. To explain this, we remind that the RSAE frequency lies close to an Alfvén continuum (AC) branch. When it approaches the AC, the radial wave numbers of traveling waves with $k_{r} > 0$ increase, $Δ k_{r} > 0$ ⁠, whereas $Δ ω < 0$ because the RSAE frequency lies above the AC (the fact the RSAE frequency decreases as the number of the radial nodes increases was observed in experiment;^9,10 it also follows from RSAE dispersion relations^23,24). As a result, $d ω / d k_{r} < 0$ ⁠, i.e., the radial group velocity $v_{g}^{t} < 0$ for waves with $k_{r} > 0$ ⁠; therefore, waves with $k_{r} < 0$ provide flux $S > 0$ ⁠. In contrast, the group velocity of traveling waves with $k_{r} > 0$ in GAEs is positive: like the RSAEs, the GAEs have frequencies close to AC, so $Δ k_{r} > 0$ ⁠, but $Δ ω > 0$ because their $ω$ lies below the AC.

The negative product $v_{g}^{t} v_{p h}^{t} < 0$ may look as a paradox, but it is actually not surprising. At the end of 1950s, it was shown that upper hybrid waves propagating along the magnetic field in the plasma cylinder have $v_{g}^{t} v_{p h}^{t} < 0$ ⁠.²⁵ The same can be the case for various cyclotron waves propagating across the magnetic field, see wave branches $ω (k)$ in, e.g., Refs. 26 and 27.

Note that because $κ S < 0$ for the RSAEs, the MT is opposite to that shown in Figs. 1 and 2: In Figs. 7 and 8, the modes rotate counterclockwise (⁠ $m < 0$ ⁠), and the energy is transmitted outward, but the mode is twisted counterclockwise, in the direction of mode rotation.

C. RSAEs in DIII-D

As an application of the findings in the previous subsection, we consider the two RSAEs observed in DIII-D, which are shown in Fig. 11 and reproduced from Figs. 3(a) and 3(b) of Ref. 10.

FIG. 11.

View large Download slide

Two RSAEs observed in the DIII-D discharge #142129. Upper panel: mode frequencies vs time, the square marks the mode without nodes and lowest characteristic radial mode number $k_{r}$ (mode 1); the triangle marks the mode with a higher $k_{r}$ (mode 2). Lower panel: the ECE cross-power vs major radius: red curve represents mode 1; blue curve, mode 2. Reproduced with permission from Figs. 3(a) and 3(b) of Heidbrink et al., Nucl. Fusion 62, 066020 (2022). Copyright 2022 IAEA.¹⁰

The modes in Fig. 11 likely represent two solutions of the eigenvalue problem: Mode 1 has a Gaussian-like radial profile, whereas mode 2 has minima and phase jumps, which probably arise from the nodes affected by the energy flux (as described in Sec. II A). This implies that a characteristic radial mode number of the second mode exceeds that of the first mode, $| k_{r 2} | > | k_{r 1} |$ ⁠. On the other hand, as seen from the figure, $Δ ω \equiv ω_{2} - ω_{1} < 0$ (we assume $ω > 0$ ⁠). This agrees with the result of Ref. 30 that RSAEs in DIII-D are anti-Sturmian (the mode frequency decreases with increasing radial mode number).

Using these facts, below we evaluate the mode group velocity as follows:

v_{g} \sim \frac{Δ ω}{Δ k_{r}} = \frac{ω_{2} - ω_{1}}{k_{r 2} - k_{r 1}} .

(59)

Because the sign of

Δ ω

is known, the sign of

v_{g}

depends on the sign of the radial mode numbers. The latter is the same as the sign of

κ

⁠.

Thus, we have to know $κ$ ⁠, i.e., the sign of the mode phase velocity $v_{p h}$ (for $ω > 0$ ⁠), which, as shown in Sec. IV B, is directed in the opposite direction than $v_{g}$ ⁠. A question, however, arises how $κ$ of our theory is related to the experimentally measured mode phase.

In the experiment, an ECE radiometer²⁸ measured the perturbed electron temperature ${\tilde{T}}_{e}$ at different radii along the outer midplane. The phase of the signal $ψ_{T}$ ⁠, defined by ${\tilde{T}}_{e} = {\hat{T}}_{e} exp (i ψ_{T})$ where ${\hat{T}}_{e} = | {\tilde{T}}_{e} |$ ⁠, was measured relative to a reference channel located near $q_{min}$ ⁠.¹⁰ For the sign convention chosen here, a positive value of $ψ_{T}$ means that the signal lags the reference signal (the opposite sign convention was employed in Fig. 3 of Ref. 10).

Below, we clarify how

ψ_{T}

is related to phases of other fluctuating magnitudes. Taking into account that the electron thermal conductivity along the field lines is very large, we proceed from the equation

B \cdot \nabla T_{e} = 0

(see Ref. 29), which after linearization takes the form

i k_{| |} {\tilde{T}}_{e} = - T_{e 0}^{'} \frac{{\tilde{B}}_{r}}{B_{0}} .

(60)

In the approximation

c \tilde{E} + \tilde{u} \times B_{0} = 0

(⁠

\tilde{u} = - i ω ξ

⁠,

ξ

the plasma displacement), we have

c E_{b} = - i ω ξ_{r} B_{0}

⁠, and the Maxwell equation

c \nabla \times \tilde{E} = i ω \tilde{B}

leads to

- c k_{| |} {\tilde{E}}_{b} = ω {\tilde{E}}_{r}

⁠. As a result, we obtain

{\hat{T}}_{e} = {\hat{ξ}}_{r} | T_{e 0}^{'} |, ψ_{ξ} = ψ_{T} + (1 + σ_{T}) π / 2,

(61)

where

σ_{T} = sgn T_{e 0}^{'}

⁠,

ψ_{ξ}

is the phase of

ξ_{r}

⁠,

ξ_{r} = {\hat{ξ}}_{r} exp (i ψ_{ξ})

⁠,

{\hat{ξ}}_{r} = | ξ_{r} |

⁠. We observe that

ψ_{ξ}

and

ψ_{T}

coincide when

T_{e 0}^{'} < 0

⁠; in the contrary case,

T_{e 0}^{'} > 0

⁠, the only difference between

ψ_{T}

and

ψ_{ξ}

is the shift by

π

⁠. One can show that, in general, phases of all fluctuation magnitudes can differ by a constant, which equals either

\pm π

or

\pm π / 2

⁠. This frequency shift has no influence on the twisting parameter

κ \equiv ψ^{'}

⁠. Due to this, our relations for twisting electromagnetic field of the modes can be applied to the ECE data.

The phase $ψ_{T}$ and corresponding twisting parameter $\bar{κ}$ of the RSAEs described by Fig. 11 are shown in Fig. 12. The experimentally measured values of $ψ_{T}$ for the two modes were interpolated with cubic splines, selected so that the obtained curves of $ψ_{T}$ are as smooth as possible while remaining within the experimental error bars. The curves of $\bar{κ}$ were obtained by differentiation of the interpolated curves of $ψ_{T}$ ⁠. We observe that $\bar{κ}$ in Fig. 12 is qualitatively similar to that in Fig. 10(b). The twisting parameter is largely negative, $κ < 0$ ⁠. Because of this, Eq. (59) leads to $v_{g} > 0$ ⁠. This implies that, as follows from Eq. (6), outward SC took place in this example. However, it should be noted that some DIII-D RSAEs have the opposite sign of twist,¹⁰ indicating that energy flows inward in those cases.

FIG. 12.

Phase ψT (upper panel) and twisting parameter κ¯ (lower panel) vs radius (r is a flux coordinate) in the same DIII-D discharge as in Fig. 11. Red, mode 1; blue, mode 2. Circles with error bars, measured values; curves, interpolation. For mode 2, 2π was added to measured values of ψT in the region r≥rmin, where rmin is the radius of the ECE power minimum which presumably represents the location of the mode zero when S(r)=0 (experimentally, ECE phase is determined modulo 2π); this was done to provide a unidirectional energy flux in the vicinity of rmin.

View large Download slide

Phase $ψ_{T}$ (upper panel) and twisting parameter $\bar{κ}$ (lower panel) vs radius (r is a flux coordinate) in the same DIII-D discharge as in Fig. 11. Red, mode 1; blue, mode 2. Circles with error bars, measured values; curves, interpolation. For mode 2, $2 π$ was added to measured values of $ψ_{T}$ in the region $r \geq r_{min}$ ⁠, where $r_{min}$ is the radius of the ECE power minimum which presumably represents the location of the mode zero when $S (r) = 0$ (experimentally, ECE phase is determined modulo $2 π$ ⁠); this was done to provide a unidirectional energy flux in the vicinity of $r_{min}$ ⁠.

Another interesting feature of the data is that the measured minima of mode 2 are an order of magnitude larger than the noise floor, consistent with our theoretical prediction that ideal nodes have non-zero amplitudes due to the energy flow. The finite spatial resolution of the ECE radiometer is partially responsible for the non-zero amplitude at the phase jumps but estimates indicate that this effect only accounts for $\sim 1 / 2$ of the observed amplitude.

V. SUMMARY AND DISCUSSION

A. Results and conclusions

Spatial channeling of the energy and concomitant mode twist during fast-ion driven instabilities are studied. Specific calculations are carried out for FMMs and two types of Alfvén eigenmodes (GAE and RSAE).

The results obtained can be summarized as follows.

It is revealed that there is a correlation between the direction of the radial wave energy flux (S), orientation of the mode twist (MT), and the direction of the mode rotation. The relations connecting the flux S and the mode twisting parameter $κ$ are obtained (for modes with one dominant poloidal harmonic, $κ = d ψ / d r$ ⁠, with $ψ$ the mode phase associated with the energy flux).

In particular, for FMMs the inward SC takes place when the mode is twisted in the direction of its rotation, whereas the outward SC occurs when the mode is twisted in the opposite direction. The same is true for GAEs. In contrast, the direction of the MT is reversed for RSAEs because, as shown in this work, the radial group velocity and radial phase velocity of traveling waves constituting the modes have opposite signs. Corresponding relations and pictures demonstrating the MT are obtained for the case when mode amplitudes are saturated. When amplitudes grow, the fluxes should be corrected. The effect of growing amplitudes is considered in a particular case when the mode damping is small compared to the drive. In general, the flux can be presented in the form $S_{s s} + S_{mode}$ as shown in Eqs. (B7) and (B8).

The modes are twisted most strongly at the radii away from the maximum mode amplitude. When the twist is small near the mode maximum, the pictures showing the mode structure may produce an illusion that the mode is not twisted at all.

The energy flux “kills” zeros in the mode amplitude, making minima instead, which are strongly twisted unless the instability drive is very strong. This effect is validated numerically in this work by observing various modes in the absence and in the presence of radial energy flux.

These findings can be used for diagnostics of the radial direction (outward or inward) of the energy transfer by the modes. In this work, two RSAEs (without and with a node) observed in a particular DIII-D discharge were considered. Using available experimental data and theoretically predicted opposite directions of the radial group velocity and phase velocity of RSAEs, it was concluded that an outward SC took place.

Observation of the MT can also be used to determine the location of the SC. Moreover, the magnitude of S can be evaluated, provided the mode amplitudes and the mode group velocity, $v_{g}$ ⁠, are known. The latter is connected with $κ$ ⁠, which can be obtained by observing and analyzing the MT. For FMM, $v_{g} = κ (r) v_{A}^{2} / ω$ ⁠. More complicated are relations for $v_{g}$ for AEs. In general, observation and analysis of the MT can supplement numerical modeling of fast-ion driven instabilities for reliable interpretation of experiments.

B. FMM vs AE

Radial energy flux is proportional to the mode group velocity,

v_{g}

⁠, see Eq. . The magnitude of

v_{g}

for FMMs and AEs can be written as

v_{g 1} = κ_{1} \frac{v_{A}^{2}}{ω_{1}},

(62)

v_{g 2} = κ_{2} \frac{v_{A}^{2}}{ω_{2}} σ,

(63)

where the subscripts “1” and “2” label quantities relevant to FMM and AE, respectively,

σ

will be specified below. It follows from here that

\frac{κ_{2}}{κ_{1}} \sim \frac{ω_{2}}{ω_{1}} \frac{1}{σ}

(64)

when

S_{2} / S_{1} \sim W_{2} / W_{1}

⁠. Then,

\frac{κ_{2}}{κ_{1}} ≫ 1

(65)

provided

σ ≪ \frac{ω_{2}}{ω_{1}} .

(66)

One can expect that $σ ≪ 1$ because the propagation of Alfvén waves across the magnetic field is weak (it vanishes for ideal Alfvén waves in a homogeneous magnetic field). If so, Eq. (64) leads to $κ_{2} ≫ κ_{1}$ ⁠, at least, for $ω_{2} \sim ω_{1}$ ⁠.

In particular, for GAEs $σ = τ k_{∥}^{2} / k_{⊥}^{2} ≪ 1$ with $τ ≪ 1$ ⁠, which results from Eqs. (53) and (54). Thus, $σ$ is really small and, therefore, the FMM twist is weaker and less noticeable than that in GAEs for reasonable ratio $ω_{2} / ω_{1}$ ⁠.

This simple consideration may explain why twists of AEs, but not of FMMs, were observed in experiments on NSTX and DIII-D. Of course, it does not mean that the FMM twist cannot be observed.

C. Mode twist in the midplane of the torus

In this work, like in previous publications (see, e.g., Refs. 9, 10, 30, and 31), the spatial structure of eigenmodes was considered in the poloidal cross section. In this plane, the relations describing the MT are different for the modes consisting of a single poloidal harmonic and the multi-harmonic modes. On the other hand, in tokamaks all kinds of eigenmodes consist of a single toroidal harmonic due to axial symmetry of these machines. Due to this, the analysis of MT in the midplane of the torus could be carried out on the same footing for any mode.

In addition, the MT in the midplane of the torus is largest when $n < m$ ⁠, which follows from Eqs. (11) and (12). In the limit case of $n = 0$ ⁠, Eq. (12) predicts $φ^{'} = \infty$ ⁠. This means that the level lines of the mode in the torus midplane are circles. This can occur during destabilization of the geodesic acoustic modes (GAMs) and other modes with $n = 0$ ⁠. For instance, a global GAM instability driven by fast ions was observed in DIII-D;^32,33 the $n = 0$ sidebands with frequencies close to those of two long-lived modes in the range $f = 500 - 600$ kHz were observed in NSTX.¹⁷

ACKNOWLEDGMENTS

This work was supported in part by the U.S. Department of Energy Grant No. DE-FG02-06ER54867 via Partner Project Agreement P786/UCI Subaward #2022-1701 between the Regents of the University of California, Irvine (UCI), the Science and Technology Center in Ukraine, and the Institute for Nuclear Research. W.W. Heidbrink was supported by the U.S. Department of Energy Grant No. DE-SC0020337.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Ya. I. Kolesnichenko: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). V. V. Lutsenko: Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal). A. V. Tykhyy: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yu. V. Yakovenko: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). W. W. Heidbrink: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX A: MODE TWIST AND ENERGY FLUX IN THE GEOMETRICAL OPTICS APPROXIMATION

We treat the eigenmode under consideration as a wave standing in the radial direction, which can be considered as a superposition of two traveling waves propagating in opposite directions,

\begin{matrix} u = Re {exp (i m ϑ - i ω t - i n φ) \times [u_{1} exp (i k_{r} r) + u_{2} exp (- i k_{r} r)]} . \end{matrix}

(A1)

Here, u is an appropriate wave function of the mode (for example, we can take

ϕ

for Alfvén modes and

B_{∥}

for FMM);

u_{1}

and

u_{2}

are the amplitudes of the traveling waves. We consider a narrow vicinity of an anti-node,

r = r_{*}

(a point where the phases of the traveling waves coincide), neglecting the radial dependence of all quantities except for the wave phase. To separate the amplitude and the phase of the mode, we present Eq. in an alternative form,

u = \hat{u} (r) Re exp [i m ϑ - i ω t - i n φ + i ψ (r)],

(A2)

where the local mode amplitude

\hat{u}

is given by

{\hat{u}}^{2} = (| u_{1} |^{2} + | u_{2} |^{2}) [{cos}^{2} (k_{r} Δ r) + α {sin}^{2} (k_{r} Δ r)],

(A3)

ψ = ψ_{*} + arctan {α tan [k_{r} Δ r]}

(A4)

is the local mode phase,

α = (| u_{1} | - | u_{2} |) / (| u_{1} | + | u_{2} |)

is the amplitude imbalance,

Δ r = r - r_{*}

⁠, and

ψ_{*}

is the phase of both running waves at

r = r_{*}

⁠.

When

u_{1} = u_{2}

⁠, the wave given by Eq. is a “pure” standing wave,

u \propto cos (k_{r} r)

⁠; otherwise, it is a mixture of a standing wave and a traveling one. Differentiating Eq. and taking

r \approx r_{*}

⁠, we find

κ = \frac{α k_{r}}{{cos}^{2} (k_{r} Δ r) + α {sin}^{2} (k_{r} Δ r)} \approx α k_{r} .

(A5)

Thus, the amplitude imbalance results in a twist of the wavefront. The direction of this twist is intuitive. It corresponds to the mode propagation in the positive-r direction when

u_{1}

dominates and vice versa.

Now, we assume that

α ≪ 1

⁠, i.e.,

u_{1} \approx u_{2}

⁠, and relate the mode twist to the energy flux due to the mode

S = v_{g}^{t} (W_{1} - W_{2}) \approx 2 α v_{g}^{t} \bar{W} .

(A6)

Here,

v_{g}^{t}

is the radial group velocity of the first traveling wave,

W_{1}

and

W_{2}

are the energy densities of the traveling waves,

\bar{W} = W_{1} + W_{2}

is the space-averaged mode energy density, and it was assumed that

W_{1, 2} \propto u_{1, 2}^{2}

⁠. Putting Eq. to Eq. , we obtain the following equation valid near the anti-node point:

S = 2 \frac{κ}{k_{r}} v_{g}^{t} \bar{W} .

(A7)

APPENDIX B: RELATION FOR ENERGY FLUX DETERMINED BY THE ENERGY BALANCE

We proceed from the following equation:⁷

\frac{\partial W}{\partial t} + \frac{1}{r} \frac{\partial}{\partial r} r S = 2 γ_{L} W .

(B1)

Integrating this equation over the radial coordinate, we obtain

\begin{matrix} r S (r) = 2 \int_{0}^{r} d r^{'} r^{'} γ_{L} W - \frac{\partial W^{(r)}}{\partial t} \\ = 2 \int_{0}^{r} d r^{'} r^{'} (γ_{L} - γ) W + 2 γ W^{(r)} - \frac{\partial W^{(r)}}{\partial t}, \end{matrix}

(B2)

where

W^{(r)} = \int_{0}^{r} d r^{'} r^{'} W

is the wave energy inside the flux surface within a certain radius r. When

W (a) = 0

⁠,

S (a) = 0

⁠. Then, it follows from Eq. that

2 \int_{0}^{a} d r^{'} r^{'} γ_{L} W = \partial W^{(a)} / \partial t

⁠. On the other hand,

2 \int_{0}^{a} d r^{'} r^{'} γ_{L} W = 2 γ W^{(a)}

⁠. Therefore,

\partial ln W^{(a)} / \partial t = 2 γ

⁠. If the mode shape is almost not affected by the instability, we can write

W (r, t) \approx A (t) f (r),

(B3)

which leads to³

\frac{\partial ln W^{(r)}}{\partial t} \approx \frac{\partial ln W^{(a)}}{\partial t} = 2 γ .

(B4)

In this case, Eq. reduces to Eq. .

Equation implies that the instability growth rate is sufficiently small,

γ W ≪ S / Δ r

⁠, where

Δ r

is the mode width. In other words,

γ ≪ \frac{v_{g}}{Δ r} .

(B5)

When Eq. is not satisfied, the initial value problem, rather than eigenvalue problem, is to be solved.

The energy flux can be presented as a sum of the flux produced in the steady state,

S_{s s}

⁠, and the flux supporting the mode shape during growth of the mode amplitude,

S_{mode}

S (r) = S_{s s} + S_{mode},

(B6)

where

S_{s s} = \frac{2}{r} \int_{0}^{r} d r^{'} r^{'} (γ_{L} + δ γ) W,

(B7)

S_{mode} = - \frac{2}{r} \int_{0}^{r} d r^{'} r^{'} (γ + δ γ) W .

(B8)

Here,

δ γ

satisfies the condition

\int_{0}^{a} d r^{'} r^{'} δ γ W = - γ W^{(a)},

(B9)

so

S_{s s} (a) = S_{mode} (a) = 0

⁠.

Note that it was assumed here that the wave field vanishes at the plasma edge. In general, a can be replaced by $\infty$ ⁠, which does not change Eq. (9).

APPENDIX C: ENERGY FLUX IN ALFVÉN EIGENMODES

When studying Alfvén eigenmodes, we proceed from the following reduced MHD equation:

\begin{matrix} \nabla \cdot (\frac{1}{v_{A}^{2}} \nabla_{⊥} \frac{\partial^{2} ϕ}{\partial t^{2}}) = B \nabla_{∥} [\frac{1}{B^{2}} \nabla \cdot (B \hat{F} ϕ)] \\ - \frac{2 π}{c} [\nabla \cdot (J \nabla_{∥} ϕ) + B \nabla_{∥} (\frac{1}{B} J \cdot \nabla ϕ)] \\ + 4 π (\frac{K}{B^{2}} Π \cdot \nabla ϕ + \frac{Π}{B^{2}} K \cdot \nabla ϕ), \end{matrix}

(C1)

where

\partial ϕ / \partial t = \tilde{Φ}

⁠,

Φ

is the mode scalar potential,

J = b \times \nabla (j_{∥} / B)

⁠,

j

is the equilibrium current,

K = b \times K

⁠,

K

is the field line curvature,

Π = b \times \nabla p

⁠, p is the plasma pressure, the operator

\hat{F}

is given by

\hat{F} ϕ = B \nabla_{⊥} (B^{- 1} \nabla_{∥} ϕ)

⁠. This equation is, in fact, Eq. of Ref. 34 with the compressibility contribution omitted. In addition, we wrote the current and pressure contributions in a form that explicitly manifests that the equation is Hermitian; this makes possible omitting either contribution without breaking this property.

Equation follows from the Lagrangian

L = K - P

(C2)

with the kinetic energy density

K = \frac{c^{2}}{8 π} \frac{1}{v_{A}^{2}} {| \nabla_{⊥} \frac{\partial ϕ}{\partial t} |}^{2}

(C3)

and the potential energy density

P = \frac{c^{2}}{8 π} {| \hat{F} ϕ |}^{2} + \frac{c}{2} (\nabla_{∥} ϕ) J \cdot \nabla ϕ - \frac{c^{2}}{2 B^{2}} (K \cdot \nabla ϕ) (Π \cdot \nabla ϕ) .

(C4)

It should be emphasized that the Lagrangian is not an abstract mathematical construct which arises due to the use of variational calculus. The energy densities

K

and

P

have real physical meaning. One can see that

K \approx ρ u_{⊥}^{2} / 2

(here

u

is the flow velocity) and the first term of

P

is

\approx {\tilde{B}}_{⊥}^{2} / (8 π)

(see expressions for

{\tilde{B}}_{⊥}

and

u_{⊥}

in Ref. 34).

We assume that the mode shape is fixed [the mode satisfies Eq. ] and proceed by Fourier decomposition, taking

ϕ

in the form

ϕ = Re \sum_{m} exp (i m ϑ - i n φ - i ω t) ϕ_{m, n} (r),

(C5)

where we choose the number of terms in the sum, depending on the mode under consideration. After straightforward calculations, one can derive the kinetic and potential energy densities averaged over the wave period and the flux surface,

\begin{matrix} \bar{K} & = \frac{| ω |^{2}}{2 V^{'}} ({\vec{ϕ}}^{H} D^{00} \vec{ϕ} + {\vec{ϕ}}^{H} D^{01} \frac{d \vec{ϕ}}{d r} \\ + \frac{d {\vec{ϕ}}^{H}}{d r} D^{10} \vec{ϕ} + \frac{d {\vec{ϕ}}^{H}}{d r} D^{11} \frac{d \vec{ϕ}}{d r}) exp (2 γ t), \end{matrix}

(C6)

\begin{matrix} \bar{P} & = \frac{1}{2 V^{'}} ({\vec{ϕ}}^{H} A^{00} \vec{ϕ} + {\vec{ϕ}}^{H} A^{01} \frac{d \vec{ϕ}}{d r} \\ + \frac{d {\vec{ϕ}}^{H}}{d r} A^{10} \vec{ϕ} + \frac{d {\vec{ϕ}}^{H}}{d r} A^{11} \frac{d \vec{ϕ}}{d r}) exp (2 γ t) . \end{matrix}

(C7)

Here, r is the radial coordinate; superscripts H denote conjugate transpose;

γ = Im (ω)

⁠; the flux-surface averaging is defined as

\bar{(\dots)} = \oint d ϑ d φ \sqrt{g} (\dots) / V^{'},

(C8)

V^{'} = \oint d ϑ d φ \sqrt{g};

(C9)

the matrices

A^{j k}

and

D^{j k}

⁠,

j, k = 0, 1

⁠, depend on the plasma configuration; they are presented in Appendix E for the case of large aspect ratio and circular cross section. One can show that the matrices

A^{00}

⁠,

A^{11}

⁠,

D^{00}

⁠, and

D^{11}

are always Hermitian,

A^{10 H} = A^{01}

⁠, and

D^{10 H} = D^{01}

⁠.

Varying the Lagrangian

\bar{L} = \bar{K} - \bar{P}

⁠, we obtain an ordinary differential equation for the vector variable

\vec{ϕ} = col (ϕ_{m_{1} n}, ϕ_{m_{2} n}, \dots, ϕ_{m_{l} n})

⁠,

ω^{2} \hat{D} \vec{ϕ} = \hat{A} \vec{ϕ}

(C10)

with the self-conjugate differential operators

\hat{D}

and

\hat{A}

given by

\hat{D} \vec{ϕ} = D^{00} \vec{ϕ} + D^{01} \frac{d \vec{ϕ}}{d r} - \frac{d}{d r} (D^{10} \vec{ϕ}) - \frac{d}{d r} (D^{11} \frac{d \vec{ϕ}}{d r}),

(C11)

\hat{A} \vec{ϕ} = A^{00} (r) \vec{ϕ} + A^{01} \frac{d \vec{ϕ}}{d r} - \frac{d}{d r} (A^{10} \vec{ϕ}) - \frac{d}{d r} (A^{11} \frac{d \vec{ϕ}}{d r}) .

(C12)

To introduce sources and sinks of the wave energy to Eq. , we add one more term to Eq. ,

ω^{2} \hat{D} \vec{ϕ} = \hat{A} \vec{ϕ} + i ω ν \vec{ϕ},

(C13)

where

ν

is a Hermitian matrix. Using Eq. , one can prove the following conservation law:

\frac{\partial \bar{W}}{\partial t} = 2 γ \bar{W} = - \frac{1}{V^{'}} \frac{d}{d r} (V^{'} \bar{S}) + 2 γ_{L} \bar{W},

(C14)

where

\bar{W} = \bar{K} + \bar{P}

is the wave energy density, the average radial energy flux (⁠

\bar{S}

⁠) is given by

\begin{matrix} \bar{S} = \frac{1}{V^{'}} Im {ω^{*} {\vec{ϕ}}^{H} [(A^{10} - ω^{2} D^{10}) \vec{ϕ} \\ + (A^{11} - ω^{2} D^{11}) \frac{d \vec{ϕ}}{d r}]} exp (2 γ t), \end{matrix}

(C15)

and

γ_{L}

is the local drive/damping rate (cf. Appendix B) given by

γ_{L} = \frac{| ω |^{2}}{2 V^{'}} \frac{{\vec{ϕ}}^{H} ν \vec{ϕ}}{W_{0}},

(C16)

where

W_{0} = \bar{W} / exp (2 γ t)

does not depend on time.

We proceed by obtaining

κ

(the radial derivative of the mode phase) for a wave given by Eq. . We write this equation in the form

ϕ = Re [\tilde{ϕ} (r, ϑ) exp (- i n φ - i ω t)],

(C17)

and present

\tilde{ϕ}

as

\tilde{ϕ} (r, ϑ) = \hat{ϕ} (r, ϑ) exp [i ψ_{Σ} (r, θ)] .

(C18)

with

\hat{ϕ} = | \tilde{ϕ} |

⁠, like it was done in Sec. II. When the wave does not consist of a single poloidal harmonic, we cannot take the mode phase in the form

ψ_{Σ} = ψ (r) + m ϑ

⁠. From

{\tilde{ϕ}}^{*} \nabla \tilde{ϕ} = \hat{ϕ} \nabla \hat{ϕ} + i {\hat{ϕ}}^{2} \nabla ψ_{Σ},

(C19)

we obtain

\nabla ψ_{Σ} = {\hat{ϕ}}^{- 2} Im ({\tilde{ϕ}}^{*} \nabla \tilde{ϕ}) .

(C20)

Finally, the radial component of this equation yields

κ = {\hat{ϕ}}^{- 2} Im \sum_{m_{1}, m_{2}} ϕ_{m_{1}, n}^{*} ϕ_{m_{2}, n}^{'} exp (i m_{2} ϑ - i m_{1} ϑ) .

(C21)

Note that

κ

depends on

ϑ

because of toroidicity.

APPENDIX D: EQUATION FOR FMM IN A PLASMA WITH FAST IONS

Maxwell equations for the perturbed electromagnetic field in the presence of fast ions can be written as follows:

\nabla \times \tilde{B} = - \frac{i ω}{c} \bar{ϵ} \tilde{E} + \frac{4 π}{c} {\tilde{j}}^{α},

(D1)

\nabla \times \tilde{E} = \frac{i ω}{c} \tilde{B},

(D2)

where

ϵ_{i j}

is the dielectric tensor of the bulk plasma,

{\tilde{j}}^{α}

is the fast-ion current. We assume that

{\tilde{E}}_{| |} = 0

and

ϵ_{11} = ϵ_{22} \equiv ϵ_{1}

⁠,

ϵ_{12} = - ϵ_{21} \equiv i ϵ_{2}

are well-known tensor components of the cold Maxwell plasma. Eliminating

{\tilde{B}}_{r}

and

{\tilde{B}}_{ϑ}

⁠, we can obtain the following equations for

{\tilde{B}}_{| |}

and components of

{\tilde{E}}_{⊥}

⁠:

k_{b} {\tilde{E}}_{r} + \frac{i}{r} \frac{d}{d r} r {\tilde{E}}_{b} + \frac{ω}{c} {\tilde{B}}_{| |} = 0,

(D3)

{\tilde{E}}_{r} = - p_{1} N_{b} {\tilde{B}}_{| |} - p_{2} \frac{c}{ω} {\tilde{B}}_{| |}^{'} - \frac{4 π}{ω} [i p_{1} {\tilde{j}}_{r}^{α} + p_{2} {\tilde{j}}_{b}^{α}],

(D4)

{\tilde{E}}_{b} = - i p_{2} N_{b} {\tilde{B}}_{| |} - i p_{1} \frac{c}{ω} {\tilde{B}}_{| |}^{'} + \frac{4 π}{ω} [p_{2} {\tilde{j}}_{r}^{α} - i p_{1} {\tilde{j}}_{b}^{α}],

(D5)

where

p_{1} = \frac{ϵ_{1} - N_{| |}^{2}}{{(ϵ_{1} - N_{| |}^{2})}^{2} - ϵ_{2}^{2}}, p_{2} = \frac{ϵ_{2}}{{(ϵ_{1} - N_{| |}^{2})}^{2} - ϵ_{2}^{2}},

(D6)

N = c k / ω

⁠, and prime denotes radial derivative. Combining Eqs. , we have

\begin{matrix} \frac{c^{2}}{ω^{2}} \frac{1}{r} \frac{d}{d r} r p_{1} \frac{d {\tilde{B}}_{| |}}{d r} + [1 - N_{b}^{2} p_{1} + \frac{c}{ω r} {(r N_{b} p_{2})}^{'}] {\tilde{B}}_{| |} \\ + \frac{4 π c}{ω^{2}} (\frac{1}{r} \frac{d}{d r} r p_{2} - k_{b} p_{1}) i {\tilde{j}}_{r}^{α} + \frac{4 π c}{ω^{2}} (\frac{1}{r} \frac{d}{d r} r p_{1} - k_{b} p_{2}) {\tilde{j}}_{b}^{α} = 0. \end{matrix}

(D7)

Using this equation, we can calculate the growth rate of an FMM instability caused by fast ions as follows. To begin, we multiply Eq. by

r {\tilde{B}}_{| |}^{*}

and integrate the product over r, which yields

\begin{matrix} \int_{0}^{\infty} d r r {- \frac{c^{2}}{ω^{2}} p_{1} | {\tilde{B}}_{| |}^{'} |^{2} \\ + [1 - p_{1} \frac{c^{2} k_{b}^{2}}{ω^{2}} + \frac{c^{2}}{ω^{2} r} {(r k_{b} p_{2})}^{'}] | {\tilde{B}}_{| |} |^{2}} \\ = \int_{0}^{\infty} d r r \frac{4 π c}{ω^{2}} {(i p_{2} {\tilde{j}}_{r}^{α} + p_{1} {\tilde{j}}_{b}^{α}) {\tilde{B}}_{| |}^{' *} \\ + k_{b} (i p_{1} {\tilde{j}}_{r}^{α} + p_{2} {\tilde{j}}_{b}^{α}) {\tilde{B}}_{| |}^{*}} . \end{matrix}

(D8)

The relation in brackets in the integrand of the RHS of this equation can be written as

{\dots} = - i (ω / c) \tilde{j} \cdot {\tilde{E}}^{*}

⁠.

To simplify this equation, we consider the limit cases of the low-frequency instabilities (⁠ $ω^{2} ≪ ω_{B i}^{2}$ ⁠) and high-frequency instabilities (⁠ $ω^{2} ≫ ω_{B i}^{2}$ ⁠). In addition, we take $N_{| |}^{2} ≪ ϵ_{1}$ ⁠, which implies that $k_{| |}^{2} v_{A}^{2} ≪ ω^{2}$ for $ω^{2} ≪ ω_{B i}^{2}$ and $k_{| |}^{2} v_{A}^{2} ≪ ω_{B i}^{2}$ for $ω^{2} ≫ ω_{B i}^{2}$ ⁠. Then, $p_{1} \approx v_{A}^{2} / c^{2}$ and $p_{2} \approx p_{1} ω / ω_{B_{i}}$ ⁠. Due to this, the term in the LHS of Eq. (D8) that is proportional $p_{2}^{'}$ is negligible when $ω ≪ ω_{B i}$ (unless the ion density gradient is very large). In the contrary case, $ω ≫ ω_{B i}$ ⁠, it can be neglected only when plasma is sufficiently homogeneous in the region of the mode location: $m L / r ≫ ω / ω_{B i}$ ⁠, where L is a characteristic length of inhomogeneity. The case of $ω = l ω_{B i}$ ⁠, with l an integer, is eliminated because effects of finite Larmor radius of the ions are neglected.

Now, we write

ω = ω_{0} + i γ

with

γ ≪ ω_{0}

⁠. This leads to

\frac{γ}{ω_{0}} = - 2 π ω_{0} \frac{Re \int_{0}^{\infty} d r r ({\tilde{j}}^{α} \cdot {\tilde{E}}^{*})}{\int_{0}^{\infty} d r r (| {\tilde{B}}_{| |}^{'} |^{2} + k_{b}^{2} | {\tilde{B}}_{| |} |^{2}) v_{A}^{2}} .

(D9)

A more compact form is

γ = - 2 π \frac{Re \int_{0}^{\infty} d r r ({\tilde{j}}^{α} \cdot {\tilde{E}}^{*})}{\int_{0}^{\infty} d r r | {\tilde{B}}_{| |} |^{2}} .

(D10)

APPENDIX E: COEFFICIENTS OF THE ALFVÉN EIGENMODE EQUATION

We present here approximate expressions for the matrix coefficients

A^{j k}

and

D^{j k}

⁠,

j, k = 0, 1

⁠, entering Eqs. and . The expressions are valid for a tokamak with large aspect ratio and quasicircular cross section. They are based on the equilibrium described in Ref. 22. We keep all terms of order

ϵ

and only those terms of order

ϵ^{2}

that are important for the RSAE existence.^21,22 Equation with these coefficients is very similar to the equations used for RSAEs in Ref. 22; the two equation systems are equivalent in the limit of

m ≫ 1

and small magnetic shear. All non-zero components of the matrices are as follows:

A_{m_{1} m_{2}}^{11} = \frac{π c^{2} R}{2} {\begin{array}{l} r k_{m_{1} n}^{2} & if m_{2} = m_{1}, \\ ε_{g 1}^{r r} r k_{m_{1} n} k_{m_{2} n} & if m_{2} = m_{1} \pm 1, \end{array}

(E1)

\begin{matrix} A_{m_{1} m_{2}}^{10} = \frac{π c^{2} R}{2} (ε_{g 1}^{r r} r k_{m_{1} n} k_{m_{2} n}^{'} \mp ε^{r ϑ} k_{m_{1} n} k_{m_{2} n} m_{2} \\ \mp ζ k_{m_{2} n} / 4 \pm m_{2} ι^{2} α / 4) if m_{2} = m_{2} \pm 1, \end{matrix}

(E2)

\begin{matrix} A_{m_{1} m_{2}}^{01} = \frac{π c^{2} R}{2} (ε_{g 1}^{r r} r k_{m_{2} n} k_{m_{1} n}^{'} \pm ε^{r ϑ} k_{m_{1} n} k_{m_{2} n} m_{1} \\ \pm ζ k_{m_{1} n} / 4 \mp m_{1} ι^{2} α / 4) if m_{2} = m_{1} \pm 1, \end{matrix}

(E3)

A_{m_{1} m_{2}}^{00} = {\begin{array}{l} \frac{π c^{2} R}{2} [k_{m_{1} n}^{2} \frac{m_{1}^{2}}{r} \\ + 2 m_{1} k_{m_{1} n} ι^{'} + m_{1}^{2} α K_{r} \frac{ι^{2}}{r}] & if m_{1} = m_{2}, \\ \frac{π c^{2} R}{2} [ε_{g 1}^{ϑ ϑ} k_{m_{1} n} k_{m_{2} n} \frac{m_{1} m_{2}}{r} \\ + ε_{g 1}^{r r} r k_{m_{2} n}^{'} k_{m_{1} n}^{'} \\ - ε^{r ϑ} m_{1} m_{2} ι ι^{'} - \frac{α ι^{2}}{2 r} m_{1} m_{2} + ζ^{'} \frac{m_{1}}{2} (k_{m_{2} n} \mp \frac{n}{2 m})] & if m_{1} = m_{2} \pm 1, \end{array}

(E4)

D_{m_{1} m_{2}}^{11} = \frac{η π c^{2} R}{2 ω_{A 0}^{2}} {\begin{array}{l} r (1 + ε_{c 0}^{r r}) & if m_{2} = m_{1}, \\ r ε_{c 1}^{r r} & if m_{2} = m_{1} \pm 1, \end{array}

(E5)

D_{m_{1} m_{2}}^{10} = \mp \frac{η π c^{2} R}{2 ω_{A 0}^{2}} ε^{r ϑ} m_{2} if m_{2} = m_{2} \pm 1,

(E6)

D_{m_{1} m_{2}}^{01} = \pm \frac{η π c^{2} R}{2 ω_{A 0}^{2}} ε^{r ϑ} m_{1} if m_{2} = m_{1} \pm 1,

(E7)

D_{m_{1} m_{2}}^{00} = \frac{η π c^{2} R}{2 ω_{A 0}^{2}} {\begin{array}{l} \frac{m_{1}^{2}}{r} (1 + ε_{c 0}^{ϑ ϑ}) & if m_{1} = m_{2}, \\ ε_{c 1}^{ϑ ϑ} \frac{m_{1} m_{2}}{r} & if m_{1} = m_{2} \pm 1. \end{array}

(E8)

Here,

ω_{A 0} = v_{A} (r = 0) / R

⁠, R is the major radius of the plasma torus,

ε = r / R

⁠,

k_{m n} = m ι - n

⁠,

ι = 1 / q

⁠,

α = - β^{'} R q^{2}

⁠,

ζ = α - ε ι

⁠,

K_{r} = α / 2 + ε (1 - ι^{2})

is the average radial field line curvature,

ε_{g 1}^{r r} = Δ^{'}

⁠,

ε_{g 1}^{ϑ ϑ} = - ε - Δ^{'}

⁠,

ε_{c 1}^{r r} = Δ^{'} + 2 ε

⁠,

ε_{c 1}^{ϑ ϑ} = ε - Δ^{'}

⁠,

ε^{r ϑ} = ε + {(r Δ^{'})}^{'}

⁠,

ε_{c 0}^{r r} = 4 ε Δ^{'}

⁠,

ε_{c 0}^{ϑ ϑ} = - 4 ε (ε + Δ^{'})

⁠,

Δ (r)

is the Shafranov shift determined from the equation

{(Δ^{'})}^{'} = ε - 2 (1 - \hat{s}) Δ^{'} + α,

(E9)

\hat{s} = r q^{'} / q

is the magnetic shear.

REFERENCES

1.

D.

Stutman

,

L.

Delgado-Aparicio

,

N.

Gorelenkov

,

M.

Finkenthal

,

E.

Fredrickson

,

S.

Kaye

,

E.

Mazzucato

, and

K.

Tritz

,

Phys. Rev. Lett

102

,

115002

(

2009

).

https://doi.org/10.1103/PhysRevLett.102.115002

Google Scholar

Crossref

PubMed

2.

Ya. I.

Kolesnichenko

,

Yu. V.

Yakovenko

, and

V. V.

Lutsenko

,

Phys. Rev. Lett.

104

,

075001

(

2010

).

https://doi.org/10.1103/PhysRevLett.104.075001

Google Scholar

Crossref

PubMed

3.

Ya. I.

Kolesnichenko

and

A. V.

Tykhyy

,

Phys. Plasmas

25

,

102507

(

2018

).

https://doi.org/10.1063/1.5048380

Google Scholar

Crossref

4.

Ya. I.

Kolesnichenko

,

A. V.

Tykhyy

, and

R. B.

White

,

Nucl. Fusion

60

,

112006

(

2020

).

https://doi.org/10.1088/1741-4326/ab8182

Google Scholar

Crossref

5.

Ya. I.

Kolesnichenko

,

Yu. V.

Yakovenko

,

V. V.

Lutsenko

,

R. B.

White

, and

A.

Weller

,

Nucl. Fusion

50

,

084011

(

2010

).

https://doi.org/10.1088/0029-5515/50/8/084011

Google Scholar

Crossref

6.

E. V.

Belova

,

N. N.

Gorelenkov

,

N. A.

Crocker

,

J. B.

Lestz

,

E. D.

Fredrickson

,

S.

Tang

, and

K.

Tritz

,

Phys. Plasmas

24

,

042505

(

2017

).

https://doi.org/10.1063/1.4979278

Google Scholar

Crossref

7.

Ya. I.

Kolesnichenko

,

V. V.

Lutsenko

,

M. H.

Tyshchenko

,

H.

Weisen

,

Yu. V.

Yakovenko

, and

J. E. T.

Contributors

,

Nucl. Fusion

58

,

076012

(

2018

).

https://doi.org/10.1088/1741-4326/aac09f

Google Scholar

Crossref

8.

Ya. I.

Kolesnichenko

,

H.-T.

Kim

,

V. V.

Lutsenko

,

A. V.

Tykhyy

,

R. B.

White

, and

Yu. V.

Yakovenko

,

J. Plasma Phys.

88

,

905880513

(

2022

).

https://doi.org/10.1017/S0022377822001039

Google Scholar

Crossref

9.

B. J.

Tobias

,

I. G. J.

Classen

,

C. W.

Domier

,

W. W.

Heidbrink

,

N. C.

Luhmann

, Jr.,

R.

Nazikian

,

H. K.

Park

,

D. A.

Spong

, and

M. A.

Van Zeeland

,

Phys. Rev. Lett.

106

,

075003

(

2011

).

https://doi.org/10.1103/PhysRevLett.106.075003

Google Scholar

Crossref

PubMed

10.

W. W.

Heidbrink

,

E. C.

Hansen

,

M. E.

Austin

,

G. J.

Kramer

, and

M. A.

Van Zeeland

,

Nucl. Fusion

62

,

066020

(

2022

).

https://doi.org/10.1088/1741-4326/ac5109

Google Scholar

Crossref

11.

R.

Ma

,

F.

Zonca

, and

L.

Chen

,

Phys. Plasmas

22

,

092501

(

2015

).

https://doi.org/10.1063/1.4929849

Google Scholar

Crossref

12.

G. J.

Kramer

,

B. J.

Tobias

,

A.

Turnbull

, and

E. M.

Bass

,

Nucl. Fusion

59

,

094001

(

2019

).

https://doi.org/10.1088/1741-4326/ab2ee3

Google Scholar

Crossref

13.

Ya. I.

Kolesnichenko

,

Yu. V.

Yakovenko

, and

M. H.

Tyshchenko

,

Phys. Plasmas

25

,

122508

(

2018

).

https://doi.org/10.1063/1.5049543

Google Scholar

Crossref

14.

O. S.

Burdo

and

Ya. I.

Kolesnichenko

,

Phys. Lett. A

384

,

126825

(

2020

).

https://doi.org/10.1016/j.physleta.2020.126825

Google Scholar

Crossref

15.

B.

Coppi

,

Phys. Lett. A

172

,

439

(

1993

).

https://doi.org/10.1016/0375-9601(93)90235-R

Google Scholar

Crossref

16.

Ya. I.

Kolesnichenko

,

V. V.

Lutsenko

,

A.

Weller

,

H.

Thomsen

,

Yu. V.

Yakovenko

,

J.

Geiger

, and

A.

Werner

,

Europhys. Lett.

85

,

25004

(

2009

).

https://doi.org/10.1209/0295-5075/85/25004

Google Scholar

Crossref

17.

N. A.

Crocker

,

E. D.

Fredrickson

,

N. N.

Gorelenkov

,

W. A.

Peebles

,

S.

Kubota

,

R. E.

Bell

,

A.

Diallo

,

B. P.

LeBlanc

,

J. E.

Menard

,

M.

Podestà

,

K.

Tritz

, and

H.

Yuh

,

Nucl. Fusion

53

,

043017

(

2013

).

https://doi.org/10.1088/0029-5515/53/4/043017

Google Scholar

Crossref

18.

K.

Appert

,

R.

Gruber

,

F.

Troyon

, and

J.

Vaclavik

,

Plasma Phys.

24

,

1147

(

1982

).

https://doi.org/10.1088/0032-1028/24/9/010

Google Scholar

Crossref

19.

S. M.

Mahajan

,

D. W.

Ross

, and

G.-L.

Chen

,

Phys. Fluids

26

,

2195

(

1983

).

https://doi.org/10.1063/1.864404

Google Scholar

Crossref

20.

H.

Kimura

,

Y.

Kusama

,

M.

Saigusa

,

G. J.

Kramer

,

K.

Tobita

,

M.

Nemoto

,

T.

Kondoh

,

T.

Nishitani

,

O. D.

Costa

,

T.

Ozeki

,

T.

Oikawa

,

S.

Moriyama

,

A.

Morioka

,

G. Y.

Fu

,

C. Z.

Cheng

, and

V. I.

Afanas'ev

,

Nucl. Fusion

38

,

1303

(

1998

).

https://doi.org/10.1088/0029-5515/38/9/304

Google Scholar

Crossref

21.

B. N.

Breizman

,

H. L.

Berk

,

M. S.

Pekker

,

S. D.

Pinches

, and

S. E.

Sharapov

,

Phys. Plasmas

10

,

3649

(

2003

).

https://doi.org/10.1063/1.1597495

Google Scholar

Crossref

22.

G. Y.

Fu

and

H. L.

Berk

,

Phys. Plasmas

13

,

052502

(

2006

).

https://doi.org/10.1063/1.2196246

Google Scholar

Crossref

23.

N. N.

Gorelenkov

,

G. J.

Kramer

, and

R.

Nazikian

,

Plasma Phys. Controlled Fusion

48

,

1255

(

2006

).

https://doi.org/10.1088/0741-3335/48/8/013

Google Scholar

Crossref

24.

H. L.

Berk

,

D. N.

Borba

,

B. N.

Breizman

,

S. D.

Pinches

, and

S. E.

Sharapov

,

Phys. Rev. Lett.

87

,

185002

(

2001

).

https://doi.org/10.1103/PhysRevLett.87.185002

Google Scholar

Crossref

25.

A. W.

Trivelpiece

and

R. W.

Gould

,

J. Appl. Phys.

30

,

1784

(

1959

).

https://doi.org/10.1063/1.1735056

Google Scholar

Crossref

26.

A. I.

Akhiezer

,

I. A.

Akhiezer

,

R. V.

Polovin

,

A. G.

Sitenko

, and

K. N.

Stepanov

,

Plasma Electrodynamics: Linear Theory

, International Series of Monographs in Natural Philosophy Vol.

1

(

Pergamon Press

,

Oxford

,

1975

).

Google Scholar

27.

Ya. I.

Kolesnichenko

and

V. V.

Lutsenko

,

Phys. Plasmas

30

,

122104

(

2023

).

https://doi.org/10.1063/5.0176395

Google Scholar

Crossref

28.

M. E.

Austin

and

J.

Lohr

,

Rev. Sci. Instrum.

74

,

1457

(

2003

).

https://doi.org/10.1063/1.1530387

Google Scholar

Crossref

29.

Ya. I.

Kolesnichenko

,

B. S.

Lepiavko

, and

Yu. V.

Yakovenko

,

Plasma Phys. Controlled Fusion

54

,

105001

(

2012

).

https://doi.org/10.1088/0741-3335/54/10/105001

Google Scholar

Crossref

30.

M. A.

Van Zeeland

,

W. W.

Heidbrink

,

R.

Nazikian

,

M. E.

Austin

,

C. Z.

Cheng

,

M. S.

Chu

,

N. N.

Gorelenkov

,

C. T.

Holcomb

,

A. W.

Hyatt

, and

G. J.

Kramer

,

Nucl. Fusion

49

,

065003

(

2009

).

https://doi.org/10.1088/0029-5515/49/6/065003

Google Scholar

Crossref

31.

N. A.

Crocker

,

W. A.

Peebles

,

S.

Kubota

,

J.

Zhang

,

R. E.

Bell

,

E. D.

Fredrickson

,

N. N.

Gorelenkov

,

B. P.

LeBlanc

,

J. E.

Menard

,

M.

Podestà

,

S. A.

Sabbagh

,

K.

Tritz

, and

H.

Yuh

,

Plasma Phys. Controlled Fusion

53

,

105001

(

2011

).

https://doi.org/10.1088/0741-3335/53/10/105001

Google Scholar

Crossref

32.

R.

Nazikian

,

G. Y.

Fu

,

M. E.

Austin

,

H. L.

Berk

,

R. V.

Budny

,

N. N.

Gorelenkov

,

W. W.

Heidbrink

,

C. T.

Holcomb

,

G. J.

Kramer

,

G. R.

McKee

,

M. A.

Makowski

,

W. M.

Solomon

,

M.

Shafer

,

E. J.

Strait

, and

M. A.

Van Zeeland

,

Phys. Rev. Lett.

101

,

185001

(

2008

).

https://doi.org/10.1103/PhysRevLett.101.185001

Google Scholar

Crossref

PubMed

33.

G. Y.

Fu

,

Phys. Rev. Lett.

101

,

185002

(

2008

).

https://doi.org/10.1103/PhysRevLett.101.185002

Google Scholar

Crossref

PubMed

34.

O. P.

Fesenyuk

,

Ya. I.

Kolesnichenko

,

H.

Wobig

, and

Yu. V.

Yakovenko

,

Phys. Plasmas

9

,

1589

(

2002

).

https://doi.org/10.1063/1.1462633

Google Scholar

Crossref

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Eigenmode twist by energy flux during fast-ion driven instabilities

I. INTRODUCTION

II. WAVE ENERGY FLUX AND MODE TWISTING: BASIC RELATIONS

A. Energy flux

B. Mode twisting (MT)

C. Persistence of the modes

III. FAST MAGNETOACOUSTIC MODES

A. A model for description of FMM instability

B. Twisting parameter $κ$

C. Specific examples

1. Zero width of drive/damping regions

2. Regions of drive and damping are well separated

3. Continuous transition from drive to damping

IV. ALFVÉN EIGENMODES

A. Global Alfvén eigenmodes

B. Reversed shear Alfvén eigenmodes

C. RSAEs in DIII-D

V. SUMMARY AND DISCUSSION

A. Results and conclusions

B. FMM vs AE

C. Mode twist in the midplane of the torus

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: MODE TWIST AND ENERGY FLUX IN THE GEOMETRICAL OPTICS APPROXIMATION

APPENDIX B: RELATION FOR ENERGY FLUX DETERMINED BY THE ENERGY BALANCE

APPENDIX C: ENERGY FLUX IN ALFVÉN EIGENMODES

APPENDIX D: EQUATION FOR FMM IN A PLASMA WITH FAST IONS

APPENDIX E: COEFFICIENTS OF THE ALFVÉN EIGENMODE EQUATION

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Eigenmode twist by energy flux during fast-ion driven instabilities

I. INTRODUCTION

II. WAVE ENERGY FLUX AND MODE TWISTING: BASIC RELATIONS

A. Energy flux

B. Mode twisting (MT)

C. Persistence of the modes

III. FAST MAGNETOACOUSTIC MODES

A. A model for description of FMM instability

B. Twisting parameter κ

C. Specific examples

1. Zero width of drive/damping regions

2. Regions of drive and damping are well separated

3. Continuous transition from drive to damping

IV. ALFVÉN EIGENMODES

A. Global Alfvén eigenmodes

B. Reversed shear Alfvén eigenmodes

C. RSAEs in DIII-D

V. SUMMARY AND DISCUSSION

A. Results and conclusions

B. FMM vs AE

C. Mode twist in the midplane of the torus

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: MODE TWIST AND ENERGY FLUX IN THE GEOMETRICAL OPTICS APPROXIMATION

APPENDIX B: RELATION FOR ENERGY FLUX DETERMINED BY THE ENERGY BALANCE

APPENDIX C: ENERGY FLUX IN ALFVÉN EIGENMODES

APPENDIX D: EQUATION FOR FMM IN A PLASMA WITH FAST IONS

APPENDIX E: COEFFICIENTS OF THE ALFVÉN EIGENMODE EQUATION

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only

B. Twisting parameter $κ$